INTEGRAL CALCULUS FOR BEGINNERS INTEGRAL CALCULUS FOR BEGINNERS WITH AN INTRODUCTION TO THE STUDY OF DIFFERENTIAL EQUATIONS BY JOSEPH EDWAEDS, M.A. FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE MACMILLAN AND CO. AND NEW YORK 1896 All rights reserved First Edition, 1890. Reprinted 1891, 1892, 1893. With additions and corrections, 1894 ; reprinted 1896. GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. PREFACE. THE present volume is intended to form a sound introduction to a study of the Integral Calculus, suitable for a student beginning the subject. Like its companion, the 'Differential! Calculus for Beginners, it does not therefore aim at completeness, but rather at the omission of all portions of the subject which are usually regarded as best left for a later reading. It will be found, however, that the ordinary pro- cesses of integration are fully treated, as also the principal methods of Rectification and Quadrature, and the calculation of the volumes and surfaces of solids of revolution. Some indication is also afforded to the student of other useful applications of the Integral Calculus, such as the general method to be employed in obtaining the position of a Centroid, or the value of a Moment of Inertia. As it seems undesirable that the path of a student in Applied Mathematics should be blocked by a want of acquaintance with the methods of solving M298720 vi PREFACE. elementary Differential Equations, and at the same time that his course should be stopped for a sys- tematic study of the subject in some complete and exhaustive treatise, a brief account has been added of the ordinary methods of solution of the more elementary forms occurring, leading up to and including all such kinds as the student is likely to meet with in his reading of Analytical Statics, Dynamics of a Particle, and the elementary parts of Rigid Dynamics. Up to the solution of the general Linear Differential Equation with Constant Coeffi- cients, the subject has been treated as fully as is consistent with the scope of the present work. The examples scattered throughout the text have been carefully made or selected to illustrate the articles which they immediately follow. A consider- able number of these examples should be worked by the student so that the several methods explained in the " book-work " may be firmly fixed in the mind before attacking the somewhat harder sets at the ends of the chapters. These are generally of a more miscellaneous character, and call for greater originality and ingenuity, though few present any considerable difficulty. A large proportion of these examples have been actually set in examinations, and the sources to which I am indebted for them are usually indicated. PREFACE. vii My acknowledgments are due in some degree to the works of many of the modern writers on the subjects treated of, but more especially to the Treatises of Bertrand and Todhunter, and to Pro- fessor Greenhill's interesting Chapter on the Integral Calculus, which the more advanced student may consult with great advantage. My thanks are due to several friends who have kindly sent me valuable suggestions with regard to the desirable scope and plan of the work. JOSEPH EDWARDS. October, 1894. CONTENTS. INTEGRAL CALCULUS. CHAPTER I. NOTATION, SUMMATION, APPLICATIONS. PAGES Determination of an Area, ...... 1 3 Integration from the Definition, 4 9 Volume of Revolution, 10 13 CHAPTER II. GENERAL METHOD. STANDARD FORMS. Fundamental Theorem, ....... 14 19 Nomenclature and Notation, ...... 20 21 General Laws obeyed by the Integrating Symbol, . . 22 Integration of x n , x~ l , 2326 Table of Results, 2628 CHAPTER III. METHOD OF SUBSTITUTION. Method of Changing the Variable, 29 32 The Hyperbolic Functions, ...... 33 36 Additional Standard Results, 3741 CONTENTS. CHAPTER IV. INTEGRATION BY PARTS. PAGES Integration "by Parts" of a Product, .... 4347 Geometrical Proof, 4849 Extension of the Rule, 5052 CHAPTER V. PARTIAL FRACTIONS. Standard Cases, ........ 55 57 General Fraction with Rational Numerator and De- nominator, 5861 CHAPTER VI. SUNDRY STANDARD METHODS. Integration of f^L ... . 6568 J \/K Powers and Products of Sines and Cosines, . . . 6974 Powers of Secants or Cosecants, ..... 75 76 Powers of Tangents or Cotangents, ..... 77 78 /rfv , etc., 7983 a + o cos x CHAPTER VII. REDUCTION FORMULAE. Integration of x m - l XP, where X = a + bx n , . . . 8789 Reduction Formulae for / x m ~ l X p dx, .... 90 93 Reduction Formulae for / sin^a; cos^a: dx, . . . 94 95 7T IT j"z r -i Evaluation of / sii\ n xdx, I ain^x cos^x dx, . . . 96 102 CONTENTS. xi CHAPTER VIII. MISCELLANEOUS METHODS. PAGES Integration of / ^).f x . 9 ....... 109117 J X. i\f Y Integration of some Special Fractional Forms, . . 118 119 General Propositions and Geometrical Illustrations, . 120 124 Some Elementary Definite Integrals, .... 125 127 Differentiation under an Integral Sign, .... 128 129 CHAPTER IX. RECTIFICATION. Rules for Curve-Tracing, ....... 135137 Formulae for Rectification and Illustrative Examples, . 13S 139 Modification for a Closed Curve, ..... 140 Arc of an Evolute, ........ 143 Intrinsic Equation, ........ 144 149 Arc of Pedal Curve, ........ 150 CHAPTER X. QUADRATURE. Cartesian Formula, ........ 153 157 Sectorial Areas. Polars, ....... 158160 Area of a Closed Curve, ....... 161163 Other Expressions, ..... '.''-. . 164165 Area between a Curve, two Radii of Curvature and the Evolute, ......... 166167 Areas of Pedals, ........ 168175 Corresponding Areas, ....... 176 177 CHAPTER XI. SURFACES AND VOLUMES OF SOLIDS OF REVOLUTION. Volumes of Revolution, ....... 183184 Surfaces of Revolution, ....... 185187 xii CONTENTS. PAGES Theorems of Pappus, ....... 188 191 Revolution of a Sectorial Area, ...... 192 CHAPTER XII. SECOND-ORDER ELEMENTS OF AREA. MISCELLANEOUS APPLICATIONS. Surface Integrals, Cartesian Element, . 195 198 Centroids ; Moments of Inertia, ..... 199 201 Surface Integrals, Polar Element, 202203 Centroids, etc., Polar Formulae, ..... 204 207 DIFFERENTIAL EQUATIONS. CHAPTER XIII. EQUATIONS OF THE FIRST ORDER. Genesis of a Differential Equation, ..... 211214 Variables Separable, 215 Linear Equations, 216 219 CHAPTER XIV. EQUATIONS OF THE FIRST ORDER (Continued}. Homogeneous Equations, 221 226 One Letter Absent, 227229 Clairaut's Form, 230233 CHAPTER XV. EQUATIONS OF THE SECOND ORDER. EXACT DIFFERENTIAL EQUATIONS. Linear Equations, 235 236 One Letter Absent, 237238 General Linear Equation. Removal of a Term, . . 239 240 Exact Differential Equations, . . . . . . 241242 CONTENTS. xiii CHAPTER XVI. LINEAR DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS. PAGES General Form of Solution, 243244 The Complementary Function, ..... 245 251 The Particular Integral, 252263 An Equation Reducible to Linear Form with Constant Coefficients, 264265 CHAPTER XVII. ORTHOGONAL TRAJECTORIES. MISCELLANEOUS EQUATIONS. Orthogonal Trajectories, 266269 Some Important Dynamical Equations, .... 270 271 Further Illustrative Examples, 272277 Answers, 278308 ABBREVIATION. To indicate the sources from which many of the examples are derived, in cases where a group of colleges have held an examination in common, the references are abbreviated as follows : (a) = St. Peter's, Pembroke, Corpus Christi, Queen's, and St. Catharine's. (j8) = Clare, Caius, Trinity Hall, and King's. (7) = Jesus, Christ's, Magdalen, Emanuel, and Sidney Sussex. (d) = Jesus, Christ's, Emanuel, and Sidney Sussex. (e) = Clare, Caius, and King's. INTEGRAL CALCULUS CHAPTER I. NOTATION, SUMMATION, APPLICATIONS. 1. Use and Aim of the Integral Calculus. The Integral Calculus is the outcome of an en- deavour to obtain some general method of finding the area of the plane space bounded by given curved lines. In the problem of the determination of such an area it is necessary to suppose this space divided up into a very large number of very small elements. We then have to form some method of obtaining the limit of the sum of all these elements when each is ultimately infinitesimally small and their number infinitely increased. It will be found that when once such a method of summation is discovered, it may be applied to other problems such as the finding of the length of a curved line, the areas of surfaces of given shape and the volumes bounded by them, the determination of moments of inertia, the positions of Centroids, etc. E. i. c. A eacn f length A, and let a and 6 be the abscissae of A and . Then b a = ?i/L Also if y = (f)(x) be the equation of the curve, the ordinates LA, QiPp $2^2* e ^ c -' through the several points L, Q v Q 2 , etc., are of lengths (a), ^(a+K), ^(a+2A), etc. Let their extremities be respectively A, P 1? P 2 , etc., and complete the rectangles AQ V PjQg, P 2 Q 3 , etc. Now the sum of these n rectangles falls short of the area sought by the sum of the n small figures, 1 , P 1 J2 2 P 2 , etc. Let each of these be supposed NOTATION, SUMMATION, APPLICATIONS. 3 .4 to slide parallel to the o>axis into a corresponding position upon the longest strip, say P n _^Q n _iMB. Their sum is then less than the area of this strip, i.e. in the limit less than an infinitesimal of the first order, for the breadth Q n .iM is h and is ultimately an infinitesimal of the first order, and the length MB is supposed finite. Hence the area required is the limit when h is zero (and therefore n infinite) of the sum of the series of n terms The sum may be denoted by a+rh = b-h a+rh=b-h S <>(a + rJi.h or where S or 2 denotes the sum between the limits indicated. Regarding a+rh as a variable x, the infinitesimal increment h may be written as to or dx. It is customary also upon taking the limit to replace the symbol S by the more convenient sign I , and the limit of the above summation when h is diminished indefinitely is then written f 6 I (x)dx] between the limits x = a and x = b" or more shortly " from a to b." b is called the " upper " or " superior limit." a is called the " lower " or " inferior limit," The sum of the n + l terms, differs from the above series merely in the addition of INTEGRAL CALCULUS. the term h(f>(a+nh) or A0(6) which vanishes when the limit is taken. Hence the limit of this series may also be written f 6 I (x)dx. a 3. Integration from the Definition. This summation may sometimes be effected by elementary means, as we now proceed to illustrate : Ex. 1. Calculate Cb / e*dx. Here we have to evaluate Lt h==Q h[e a + e a+h + e a+ + . . . where b = a + nh. This =Lt h ^h^p\e a =Lt h ^(e b - e a )-^-=e* - e\ & 1 & X [By Diff. Calc. for Beginners, Art. 15.] /b r=n-l xdx we have to find Lt 2 (+rA)A, where r= Now 2(a + rh)h = and in the limit becomes 2 22' / 6 1 $x we have to obtain the limit when h is a indefinitely diminished of NOTATION, SUMMATION, APPLICATIONS. 5 > a b + h a-h &' and when h diminishes without limit, each of these becomes II a b' Thus f*JL /&==*.* J .r 2 a b a Ex. 4. Prove ab initio that /& sin # ofo? = cos a - cos 6. We now are to find the limit of [sin a + sin(a + k) + sin(a + 2A) + . . . to n terms]A, sinf a+n l- Jsin n- \ 2/ 2, sin | * This expression = cosf a - J cos < a + (2n - 1)- j- - 2JJ sin- 2 sm- which when A is indefinitely small ultimately takes the form cos a cos b. INTEGRAL CALCULUS. EXAMPLES. Prove by summation that / sir 2. / sinh xdx cosh b cosh a. 3. /b cos OdO = sin 6 sin a. 4. Integration of x m . As a further example we next propose to consider the limit of the sum of the series h[a m + (a + h} m + (a + 2h) m +. i 7 6 a where h = -- , n and n is made indefinitely large, m + 1 not being zero. fy I I\m+l _ yin + 1 [Lemma. The Limit of v>/ % - 2 is m + 1 when A is Ay m indefinitely diminished, whatever y may be, provided it be of finite magnitude. For the expression may be written - 1 y and since h is to be ultimately zero we may consider - to be y less than unity, and we may therefore apply the Binomial / ^\7?l + l Theorem to expand ( 1 -J-- J , whatever be the value of m+l. NOTATION, SUMMATION, APPLICATIONS. 7 (See Dif. Gale, for Beginners, Art. 13.) Thus the expression "becomes -x(a convergent series) y m + I when A is indefinitely diminished.] In the result put i/ success! vely = a, a+h, a+2h,etc....a + (n l)h, and we get l -a m + l _ T ( ~ _ r , _ 1 - (a + n^ h(a+n-Ui) m or adding numerators for a new numerator and de- nominators for a new denominator, fe[a w + (a + /t) w + (a + 2h) m + . . . + (a + n^l or Lt h=Q h[a m + (a + A) m + (a m+1 ' In accordance with the notation of Art. 2, this may be written ' 6 7 b x m dx= m+1 8 INTEGRAL CALCULUS. The letters a and b may represent any finite quantities what- ever, provided x m does not become infinite between x=a and When a is taken as exceedingly small and ultimately zero, it is necessary in the proof to suppose h an infinitesimal of higher order, for it has been assumed that in the limit - is zero for all the values given to y. V When 6 = 1 and a = 0, ultimately the theorem be- comes x m dx= 7 if m + 1 be positive, o or = oo if m + 1 be negative. This theorem may be written also r according as m+1 is positive or negative. The limit or, which is the same thing, -Lst n= oo M4-' differs from the former by , i.e. by in the limit, 1 n and is therefore also -?, or oo according as m+1 is positive or negative. The case when m + l=0 will be discussed later. Ex. 1. Find the area of the portion of the parabola 7/ 2 =4a# bounded by the curve, the #-axis, and the ordinate xc. NOTATION, SUMMATION, APPLICATIONS. Let us divide the length c into n equal portions of which NM is the (r+l) th , and erect ordinates NP, MQ. Then if PR be drawn parallel to NM, the area required is the limit when n is infinite of the sum of such rectangles as PM (Art. 2), i.e. Lt^PN.NM or where nh = c. Now [By Art. 4.] Area =f =f of the rectangle of which the extreme ordinate arid abscissa of the area are adjacent sides. Ex. 2. Find the mass of a rod whose density varies as the with power of the distance from one end. Let a be the length of the rod, o> its sectional area supposed uniform. Divide the rod into n elementary portions each of length -. The volume of the (r+l)th element from the end of zero densitv is w-, and its density varies from ( | to (7+la\ m n * *" 1 - ) . Its mass is therefore intermediate between \ n ) coa** 1 - and ** 10 INTEGRAL CALCULUS. Thus the mass of the whole rod lies between and and in the limit, when n increases indefinitely, becomes ra+1 5. Determination of a Volume of Revolution. Let it be required to find the volume formed by the revolution of a given curve AB about an axis in its own plane which it does not cut. Taking the axis of revolution as the cc-axis, the figure may be described exactly as in Art. 2. The Fig. 3. elementary rectangles AQ V P-fy^ P 2 Qz> etc., trace in their revolution circular discs of equal thickness, and of volumes (a + 2h)+... + 0(6 - h) + 0(6)], when h is diminished indefinitely, and therefore n increased without limit. [It may at once be seen that this limit is finite, for if <$>(a+rh) be the greatest term the sum is - a)(a + rh) + h<$>(a + GENERAL METHOD. STANDARD FORMS. 15 which is finite, since by hypothesis <(#) is finite four all values of x intermediate between b and a.] Let \fs(x) be another function of x such that (a etc., Ih) = \[s(a + nil) \ where the quantities a 2 , a 3 , ..., a n are all, like a v quantities whose limits are zero when h diminishes indefinitely. By addition, h[(a) + 0(a + h) + (a Let a be the greatest of the quantities a v a 2 , . . . , a n , then Afoi+ag+^.+On] is (6) is in the limit zero; hence if we desire, it may be added to the left-hand member of this result, and it may then be stated that .e. 1 (x) is the differential coefficient) is obtained, the process of algebraic or rb trigonometric summation to obtain I (x)dx express the limit when a b becomes infinitely large of ^(6) ^js(a), i.e. I (j)(x)dx = Lt b=x I (x)dx. Ex. 1. The differential coefficient of ^ - is plainly x m . Hence if <$>(x)=x m we have df(x):L- and / x ^ ' m+l J m+1 m + \ m+l GENERAL METHOD. STANDARD FORMS. 17 Ex. 2. The quantity whose differential coefficient is cos a? is known to be sin x. Hence 6 cos x dx = sin b sin a. Ex. 3. The quantity whose differential coefficient is e* is itself e x . Hence Ex. 4. EXAMPLES. Write down the values of 1. /Vdr, 2. /Vcfo?, 3. cfo, 4. /b rl ,-2 X CiX) 2i, I X Cf/JCm o. I X d/X^ a 1 it ir /2 rA r4 cos x dx, 6. / sec 2 ^; dx^ 7. / \ o ia 8. Geometrical Illustration of Proof. The proof of the above theorem may be interpreted geo- metrically thus : Let AB be a portion of a curve of which the ordinate is finite and continuous at all points between A and B, as also the tangent of the angle which the tangent to the curve makes with the a?-axis. Let the abscissae of A and B be a and b respectively. Draw ordinates A N, BM. Let the portion NM be divided into n equal portions each of length h. Erect ordinates at each of these points of division cutting the curve in P, Q, R, ..., etc. Draw the successive tangents AP^ PQi, QRi, etc., and the lines AP 2 ,PQ 2J QR 2 ,..., parallel to the ,r-axis, and let the equation of the curve be y = ^r(x\ and let V^') = then (a\ <$>(a + h\ <$>(a + Zh\ etc., are respectively tanP.JPj, taii^Pft, etc., E. I. C. B 18 and INTEGRAL CALCULUS. -h), ..., are respectively the lengths Now it is clear that the algebraic sum of P 2 P, 2 , R 2 R, ..., is MB-NA, i.e. Hence u s, L Fig. 6. M x Now the portion within square brackets may be shewn to diminish indefinitely with h. For if R^ for instance be the greatest of the several quantities PjP, Q^ etc., the sum [P 1 P+Q 1 Q+...] is (x) between limits a and b ; fx I (x)dx or -^(x) without any specified limits and regarded merely as the reversal of an operation of the differential calculus is called an "indefinite" or " unconnected " integral. 12. Addition of a Constant. It will be obvious that if (x) + any arbitrary constant. (2) The operation of integration is distributive; for if u, v, w be any functions of x, -T-j |u^+l^^+l^^r and therefore (omitting constants) JurZ#+ |i;cfe-f \wdx = l (3) The operation of integration is commutative with regard to constants. (I'll For if -j = v, and a be any constant, we have d , du so that (omitting any constant of integration) au = \av dx 9 or a\vdx=\avdx, which establishes the theorem. GENERAL METHOD. STANDARD FORMS. 23 16. We now proceed to a detailed consideration of several elementary special forms of functions. 17. Integration of x n . By differentiation of - ^ we obtain n + 1 d x n+l _ _ - nrfll dx n + l Hence (as has been already seen in Art. 4 and in Art. 7, Ex. 1) Thus the rule for the integration of any constant power of x is, Increase the index by unity and divide by the index so increased. For example, /nA r 5 Q 11 r x~^ 1 X X = ~'' } X x= if ; } x T4 = "4^4- EXAMPLES. TTn'^e down the integrals of 1. x 1 #" ^7 999 # 1000 . O ^"J } -^"Jf b c a O. 24 INTEGRAL CALCULUS. 18. The Case of x~\ It will be remembered that x~ l or - is the differ- x ential coefficient of log x. Thus fl \-dx = logx. Jx This therefore forms an apparent exception to the general rule f ^n+l \x n dx = 19. The result, however, may be deduced as a limiting case. Supplying the arbitrary constant, we have /x n dx = + C = ~ -I- A n+l n+l where A = C+ . n + l and is still an arbitrary constant. Taking the limit when n + l=0, - takes the form log x, - [Diff. Calc. for Beginners, Art. 15.] and as C is arbitrary we may suppose that it contains a nega- tively infinite portion - - - together with another arbitrary 7i ~\~ J. portion A. Thus Lt n== -i {x n dx = log x + A. 20. In the same way as in the integration of x n we have 1 = (n + V)a(ax + b) n and ^-log(a% + 6) = &v GENERAL METHOD. STANDARD FORMS. 25 f/ IA 7 (oo5 + &) n+1 and therefore \(ax+o) n dx = / , ' y (n+l)a and I 'r = -\og(ax + b\ Jax + b a 6V f 1 fFor convenience we shall often find jdx Jax + b printed as -r, I , dx as I ,- o , etc.] Jax + b' Jja 2 +x* J*Ja* + x* EXAMPLES. Write down the integrals of 1. ax, of 1 , a+x, a x, ax? 2 a x_ a+x 1 x a? x ' a-\-x 3. a + x a bx (a-#) 2> (a x) n * a+x a-x* x+a x d (a+x) 2 (a xY 21. We may next remark that since the differential coefficients of [(x)] n+l and of log $(x) are respectively and we have {[(x)] n '(x)dx = and The second of these results especially is of great use. It may be put into words thus : the integral of any fraction of which the numerator is the differential coefficient of the denominator is log (denominator). For example, INTEGRAL CALCULUS. /co\,xdx = / . *" x dx = log sin #, J Sill X /tan .# dx = I -a?^ = log cos x = log sec x- J cosx EXAMPLES. Write down the integrals of n e x , ~-, (a ~\~ Ct 22. It will now be perceived that the operations of the Integral Calculus are of a tentative nature, and that success in integration depends upon a know- ledge of the results of differentiating the simple functions. It is therefore necessary to learn the table of standard forms which is now appended. It is practically the same list as that already learnt for differentiation, and the proofs of these results lie in differentiating the right hand members of the several results. The list will be gradually extended and a supplementary list given later. PRELIMINARY TABLE OF RESULTS TO BE COMMITTED TO MEMORY. GENERAL METHOD. STANDARD FORMS. r,X 27 tUj U/^ (e x dx 1 cos x dx \ = sin x. 1 sin x dx = cos#. 1 sec?x dx = tan x. Icosec 2 jc? x = cot x. \ silix dx = sec x. [ GOSX dx cosec 33. Jsin x 1 tan x dx 1 cot x dx = log sec x. \ Art. 21. = log sin 03. 1 f dx sin ~ ^ or ~~ cos ~ * t dx a 1 T 1 X . -I tC/ JL a a sec' 1 f dx a a a 1 ,05 1 QOP "" OT* i - OC\ Jx^/x 2 a ? a a a f dx vers ~" ^ or cov ers" 1 -* a ) j*/2ax i 2 a 24. It is a help to the memory to observe that all those integrals of the above list which begin with the letters "co," as cos X) cos" 1 -, covers" 1 -, etc., have a negative sign prefixed to 28 INTEGRAL CALCULUS. them. The reason is obvious. Each of these functions decreases as x increases through the first quadrant ; their differential co- efficients are therefore negative. Also it is a further help to observe the dimensions of each side. For instance, x and a being supposed linear, / is of zero i v a X" 1 dimensions. There could therefore be no - prefixed to the in- C dx a tegral. Again / 2 2 is of dimensions -1. Hence the result of J d~" -\- X integration must be of dimensions -1. Thus the integral could not be tan" 1 - (which is of zero dimensions). The student should a therefore have no difficulty in remembering in which cases the factor - is to be prefixed. a EXAMPLES. Write down the indefinite integrals of the following func- tions : ' 2. 3. * cos 2 -, cos s # . sin #, 2 4. cot x + tan x, cos^f . -+-^- V \sin^7 snr^/ # e + e* ' log sin x ^sec- 1 ^. \/^-i* CHAPTER III. METHOD OF SUBSTITUTION. 25. Change of the Independent Variable. The independent variable may be changed from x to z by the change x = F(z), by the formula V being any function of x. Or if we write V=f(x), the formula will be To prove this, it is only necessary to write u = \Vdx\ then =F. du du dx ~r r dx But -- = -_-= V-j- 9 dz dx dz dz whence u = I J -j-. dz 30 INTEGRAL CALCULUS. Thus to integrate / -dx, let tan~ 1 #=;s. Then 1 dx n and the integral becomes * dz 26. In using the formula after choosing the form of the transformation x = F(z\ it is usual to make use of differentials, writing the equation j^=F'(z) as dx = F'(z)dz] the formula will then be reproduced by replacing dx of the left hand side by F'(z)dz, and x by F(z). Thus in the preceding example, after putting tan~ 1 ^t i =0, we may write *=d* and - I+x* l+.r 27. We next consider the case when the integration is a definite one between specified limits. The result obtained above, when x = F(z) is Let then and if the limits for a; be a and b, we have METHOD OF SUBSTITUTION. 31 Now when x = a, z = F~ \a) ; and when x = b, z = F~ \b}. Also f{F(z)}=-j^,{F(z)}, and whence so that the result of integrating f{F(z)}F'(z) with re- gard to z between limits F~\a) and F~\b) is identical with that of integrating f(x) with regard to x between the limits a and 6. Ex.1. Evaluate / - cos \Txdx. J N /a? Let xz^^ and therefore dx=Zzdz ; - cosfjxdx I -cos2. 2-2(^2=2 / cos z dz = 2 smz ^x J z J Ex. 2. Evaluate / .Aos x^dx. Let ^ 3 =2;, and therefore 3xPdx=dz ; /. / ^ 2 cos x*dx = llcoszdz = ^smz=^ sin x\ Put ^?=tan e, then dx=sec 2 0d6 ; when # = 0, we have = 0, when # = 1, we have 0=. T ; 4 32 INTEGRAL CALCULUS. ir ir :. f -T^=- dx = P *?| sec 2 dB = fsec B tan dB \J\+* { sec# { = fsec = sec - - sec = V2 - 1. Ex.4. Evaluate f x dx _ x [i.e. \ Tsech^^]. 4 e ' j o Let 6^ = ^, then e x dx = dz. When #=0, 0=1, and when # 3=e. Hence = rtan-'/V= ten- > - tan-1 = t 2 L Ji o The indefinite integral is tan~V. EXAMPLES. 1. Integrate e x cose x (Put ^=4 - cos(log x) (Pat log x = 4 x 2. Evaluate \-=-,dx (Put x*=z\ I* (Put a*=z). J 1+^ 4 " J l-f# 6 v reintegrate acos# + - -, a^sin e x + b tanh x. Evaluate f l ^ a (Put ^+1=4 Q 5. Evaluate /" dx - (Put 6. Evaluate / a * (Put x-\=z). 7. Evaluate /* da? (Put ^=0 2 ). J 2V^(1+^) 8. Evaluate [ - 1 dx. 9. Evaluate / dx. J 2W# - 1 METHOD OF SUBSTITUTION. 33 NOTE ON THE HYPERBOLIC FUNCTIONS. 28. Definitions. For purposes of integration it is desirable that the student shall be familiar with the definitions and fundamental properties of the direct and inverse hyperbolic functions. By analogy with the exponential values of the sine, cosine, tangent, etc., the exponential functions & e -* e x +e~ x e x -e~ x _ _ __ PTP 2 ~ 2 e?+e-*' are respectively written cosh x, tanh#, etc. 29. Elementary Properties. We clearly have -, C/ ~ V CJLJ.iJ.l X tanh x = - coth x = e~ x sinho? t&rihx -=cosh ^ 2 sinh x cosh x = 2 : ^ . -^ = ^ = sinh2#, AM with many other results analogous to the common formulae of Trigonometry. E. i. c. c 34 INTEGRAL CALCULUS. 30. Inverse Forms. Let us search for the meaning of the inverse function sinh" 1 ^. Put t then x = smh y = and & = x Thus y = log(x and we shall take this expression with a positive sign, viz., log (& + >v/l + # 2 ) as sinh" 1 ^. 31. Similarly, putting cosh~ 1 x = y, we have ty+e-y x cosh y == JL and e*y and ey = x*Jx*-l, whence y = log (x *Jx l 1), and we shall take this expression with a positive sign, viz., /# 2 1 as 32. Again, putting i&nh- l x y, we have x = tanh y = - and therefore e 2 y=- 1 x whence tanh - l x = 4- log ^ - " S 1 05 METHOD OF SUBSTITUTION. 35 Similarly coth ~ l x = J log --- r-. X JL 33. We shall thus consider . , ,x .,, , sinh" 1 - synonymous with log cosh" 1 - synonymous with log" X Ct"^~X tanh " l - synonymous with J log , and coth" 1 - synonymous with ^ log C "" Cfc 34. The Gudermannian. Again, the function cos^sechu is called the Guder- mannian of u and written If ^ = c sin x \/l sech 2 u = tanh it, a tanh u . , and tan x = -, = sinh u. seen u Hence gd u = cos ~ 1 sech u sin " Hanh u = tan ~ x sinh u. 35. Further, if u = log tan( -r + ^ ), \Qf 2t/ l + tan| we have e u = ; 36 INTEGRAL CALCULUS. x c u -l whence tan- = , x e u -l 2 tan 2 ^TTT and tan a; = = 2 .e- 0_ e - ! ~4^- : ~2~ Hence ^ = tan ~ 1 sinh u = gd u. Thus logtan(j+|)=gd-^ the inverse Gudermannian of x. EXAMPLES. Establish the following results : 1. /cosh#cfo?=sinh#. 4. /cosech 2 .rc^= -coth#. 2. / sinh xdx = cosh x. 5. / sm , .^ o?.r = sech x. J J cosh% 3. (sech 2 A'dr=taiihtf. 6. J 7. Writing sg x for sin gd x, etc., establish the following results : (a) / METHOD OF SUBSTITUTION. 37 ,* 36. Integrals of and X -4- f The differential coefficient of log e - is * . , 1 = log = smh - l - . z a a . ., , F dx , x + \/x 2 a? , ^ bimilarly .. = log -- =cosh -1 -. Jx/^ 2 -^ 2 a a 37. In the inverse hyperbolic forms these results resemble that for the integral I , ^, viz., sin' 1 -* Va 2 -* 2 a and the analogy is an aid to the memory. 38. We might have established the results thus : f dx To find I t =. put oj = a sinh it, then )*Jx*+a^ dx = a cosh u du and \/x 2 + a z = a cosh u. Hence = leZu = t6 = sinh" 1 -. W^ 2 + tt 2 J a Similarly putting a? = a cosh u, we have f dx Fa sinh u du f 7 11^ ! / a = I- ri - = laK =u = cosh~ 1 -. Jx/x 2 a 2 J sinh u J a Integrals of 39. To integrate *A 2 -tf 2 . Let a? = a sin ; then dx = a cos d$, 38 and INTEGRAL CALCULUS. {+/tf-^dx = ia sin 6 . a cos 0+ -^ or - sin- a 40. To integrate Let cc = a sinh z, then , cf^ = acosh0 then since 1 + sinh 2 = cosh 2 z, we have I J^^dx = a 2 \ ..... . Va 2 cosh 2 z dz = |a sinh . a cosh "z + -^- . , 2-smh- 1 - x METHOD OF SUBSTITUTION. 39 41. To integrate Let x a cosh z, then dx = a sinh z dz ; then since cosh% 1 = sinh 2 0, J Jsif^cPdx = a 2 j sinh 2 dz C62? = Ja sinh z . a cosh ^-, \ 2 a a 2 ! or -log- 42. If we put tan# = , and therefore we have __ [by Art. 40.] tan x sec 05 , - , ., ^ h J log(tan a? + sec ^), snce n -, or _ - _ + J log^ 2 cos 2 # & 1 40 INTEGRAL CALCULUS. 43. Integrals of cosec x and sec x. Let tan ^ = z ; taking the logarithmic differential 2t 1 9 x 7 dz dx dz ^dx = or - ^ -; . n , x 2 z smx z 2tan 2 Thus I cosec xdx\ = log z = log tan ^. In this example let x = -= + y. 2* Then dx = dy, and sec ydy = log tan (T + 9)- Hence I sec xdx = log tan ( r+s) or & ~ * x - 44. We have now the ADDITIONAL STANDARD FORMS, dx , x+\/x*+a? f dx . Jx/^+o 1 g f dx JTP^l =1 g \\/a 2 x 2 dx = I +/x 2 4 a 2 die = l+Jx 2 a 2 dx = a ,x f i 2 2 G a METHOD OF SUBSTITUTION. I cosec x dx = log tan^. Isecsccfo =log tanf ^- EXAMPLES. Write down the integrals of x x 3. 4. 5. 7. 8. cosec 2#, cosec(a#+&), 1 -sin 2 ^' 1 tanV 3sin^- ^' a sin ^7+ 6 cos x 10. Deduce / cosec#cfo?=logtan - by expressing cosec x as / ^j 11. Find \SQexdx by putting sin x=z. 42 INTEGRAL CALCULUS. / 12. Show that / sec x dx = cosh 13. Integrate 1 tflogtf' when l r x represents log log log ... ^7, the log being repeated r times. 15. Prove [ST. PETER'S COLL., etc., 1882.] CHAPTER IV. INTEGEATION BY PARTS. 45, Integration " by Parts " of a Product d , dv t du Since - (uv} = u - +v -, it follows that uv = I u^- dx + 1 v-j- dx, t dv , f du 7 or lu-j-dx = wu \ v-j~ dx. J dx j dx 7 / If u = (x) and ^- =\{s(x), so that ^=jT/r(#)cfe, the above rule may be written \<}>(x)\ls(x)dx= $(x)\ \\ls(x)dx\ - 0'(oO| J^H[aj)da5 Jcte; or interchanging ^(35) and V r ( a; )> J^(oj)^(a?)dte=^(aj) I (x)dx j^'(^) ' : J0(a?)efcB \dx. Thus in integrating the product of two functions, if the integral be not at once obtainable, it is possible when the integral of either one is known, say ^(x), to connect the integral \(x)\ls(x)dx 44 INTEGRAL CALCULUS. dx with a new integral 0'(#) lx/r(#)cfo \ which may be more easily integrable than the original product. 46. The rule may be put into words thus : Integral of the product (x)\{s(x) = 1st function x Integral of 2nd -the Integral of [Diff. Co. of 1st x Int. of 2nd]. Ex. 1. Integrate x cos nx. Here it is important to connect if possible j xcosnxdx with another integral in which the factor x has been removed. This may be done if x be chosen as the function $(x\ since in the second integral <$(x\ i.e. unity, occurs in place of x. Then Thus by the rule (x vxnxd**,*!*?- /"l.5 J ' n J sin 9^7 If cosn&N x - "~~~~\ - I n n\ n / sin nx cos nx ' 47. Unity may be taken as one of the factors to aid an integration. Thus / log xdx / 1 . log x dx = x log x / x -(log x)dx =x log x I \dx INTEGRA TION B T PARTS. 45 48. The operation of integrating by parts may be repeated several times. mt. f 9 ? # 2 sin nx f sin nx 7 Thus / # 2 cos nx dx - / 2# - dx. J n J n and finally, f J n / Hence $ x^ nx dx =**** -*\ _^ COS J n nL n _ # 2 sin nx Zx cos nx 2 sin nx ~ I T9 7^ * 49. If one of the subsidiary integrals returns into the original form this fact may be utilized to infer the result of the integration. Ex. 1. / e ax sin bx dx = sin bx-~\ e^cos bx dx, and / e ax cos bx dx = cos bx+-l e^sin bx dx ; therefore, if P and Q stand respectively for / e ax $m bx dx and / e ax cos bx dx, we have aP +bQ = e ax sin bx, and -bP+aQ=ef tx co8bx > n nr ct sin bx b cos bx whence P=e ax - = r? a 2 +o 2 / and w+v (a 2 + 6 2 )~ Y e ax cos ( bx tan" 1 - ). \ aJ 46 INTEGRAL CALCUL US. The student will observe that these results are the same that we should obtain by putting n= I in the formulae ^)>& ss < i t^<^(^+^)' [Diff. Gale, for Beginners, Art. 61, Ex. 4.] And this is otherwise obvious. For if to differentiate rx sm /j^,\ j g fag game as to multiply by a factor Va 2 + 6 2 and to pnsi^ / r > > increase the angle by tan" 1 -, the integration, which is the a inverse operation, must divide out again the factor Va 2 +6 2 and diminish the angle by tan" 1 -. Ex. 2. Integrate \/a 2 x l by the rule of integration by parts. r , _ _ J A/o2^ 2 ^= [Note this step.] % c a 2 sin~ l - i - I v a 2 CL J whence, transposing and dividing by 2, which agrees with the result of Art. 39. Ex. 3 Integrate e* x sm 2 x cos 3 ^. Here e 3x sin% cos 3 # = (1 cos 4#)cos x -= _(2e 3a: cos -x - INTEGRA TION B Y PAR TS. 47 Hence, by Ex. 1, ^ I e^siiA cos 3 # dx \ -j = cos f x tan~ 1 r j - J_ cos(3^-^--^cosf5^-tan- 1 |)l 3\/2 ,: V 4/ ^34 V 3/J [Compare Ex. 16, p. 55, Diff. Gale, for Beginners, putting n= l in the result.] EXAMPLES. Integrate by parts : 1. xe x , x^e*, x z e?) x cosh #, 2. ^?COS^7, ^ 2 COS07, ^?COS2 3. x sin x cos #, ^ sin x sin 2# sin 3^. 4. # 2 logtf, ^ n log^7, ^ n (log^) 2 . 5. e^sin^costf, e*sin x cos ^ cos 2#. 6. e ax sin^ sin qx sin r^?. 7. Calculate |^sin^^, / *x sin 2 ^p c?^ 1 , / 8. Show that 9. Integrate Isin" 1 ^^, /^sin" 1 ^^, \ 50. Geometrical Illustration. Let PQ be any arc of a curve referred to rectangular axes Ox, Oy, and let the coordinates of P be (X Q , y Q ), and of Q (x v y^). Let PN, QM be the ordinates and PN V QM 1 the abscissae of the points P, Q. Then plainly area PNMQ = rect. OQ - rect. OP - area But area PNMQ = f 48 and Thus INTEGRAL CALCULUS. area PN^M^Q = I x dy. ri cv\ o o J y M, N, f & y r^ -^r jydx x o A o N M x Fig. 7. Let us now consider the curve to be defined by the equations and y and let t and t be the values of t corresponding to the values # , y , and a^, 2/1 of cc and y respectively. We then have and and ri 7 ri , r*i I 2/S n I sin nxdx= , J n E. I. C. D 50 INTEGRAL CALCULUS. xv f 7 sinrac G T = I cos nx dx = , at. 7 eosnx , sinnx o\ = Ice sin nx dx = x \ ~ ^ = \x cos T^CC ax = ~ t = \ J all others can be deduced by successive applications of the above formulae. 52. Extension of the Rule for Integration by Parts. If u and v be functions of x and dashes denote differentiations and suffixes integrations with respect to x we may prove the following extension of the rule for integration by parts, uvdx = uv l u'v \ where u^ n ~ 1 ^ is written for u with TI 1 dashes; for \uvdx =uv l \u\dx, Vufv^dx =u'v 2 \u rf v^dx, \vtf'v 2 dx =u"vz Vuf'Vzdx, I u'"v B dx = ufv^ I u^'v^dx, etc. = etc. I u (n - l) Vn _ 1 dx = u( n ~ Vv n - I vf^V n dx. INTEGRA TION B Y PARTS. 51 Hence adding and subtracting alternately * 1 u v dx = uv l u'v 2 + u"v s u'"v4 +... Ex. 1. If we apply this rule to \x m e ax dx, we immediately obtain (x m e a *dx=x m -mx m - l J a a 2 Ex. 2. It will be at once seen that the integrals / # m sin nx dx and / ^ m cos nx dx of the last article may be treated in this way. EXAMPLES. "Write down the integrals of 1. x*e*, ^coshtf, ^sinh^. 2. .r 2 sin a, ^? 3 sin #, ^ 3 sin 2 ^;, ^ 4 sin x cos x. 3. Evaluate / ^sin^o?^, / x^co^xdx^ I *0 53. The determination of the integrals l# n e aa; sin bx dx, Ja3 n e aa; cos bx dx, may be at once effected. For remembering n where r = va 2 + 6 2 and tand>=-, ct/ 52 INTEGRAL CALCULUS. we have I x n e ax sm bx dx = e ax sin (bx d>) ^ e ax sin (bx J r r 2 r 3 ^^~ n\ or e ax {P sin bx Q cos where X - 3- COS 30 ... x n x n ~ l x n ~^ Q= sin n ^- sin 20 + n(n 1) ^- sin 30 ... Similarly L^a* cos i x dx = eP*{P cos bx+Q sin bx}. Ex. 1. Integrate ix^smxdx. Since \e*smxdx S^e^sinf .r -^J, we have f^ 3 e a: sm^^=^ 3 2'^e a; sm('.r - ^ - 3^ 2 2~Vsin^ - . 2" Vsinf .77 - ? V- 6 . 2~VsinCr- TT) \ 4 / =etc. Ex. 2. Prove /r=n ^| - r jQ ^Vto^ito-^-iy^j^^ ^s EXAMPLES. 1. Integrate (a) fe mai ~ lx dx. (d) /" (5) (sfaitr^xdx. (e) \ (c) Ixv&Pxdx. '(/) /"cos- 1 ^. INTEGRA TION B Y PARTS. 53 2. Integrate (a) [ x sm " 1 f dx. (c) /sin- 1 ' (&) /^5^&'. (d) /ptn tan - l x r pin tan - l x ~ -dx. (c) dx. J *-* dx. 4. Integrate ../*.. ... . .. r (a) I e(suix + cosx)ax. (a) \x (b) I xefsm^x dx. (e) I ^ 2 2*sin 2. dx. (c) I cosh ax sin bx dx, (/) / cos -j b log \dx. 5. Integrate / log - sin' 1 ^ dx. J x 6. Integrate 7. Integrate 8. Integrate /cos 201og(l+tan 0)dO. 9. Integrate (d) J*4|^. ^^ TKJPOS) 1892<] (&) 1-cos^ [a, 1892.] i /\ T- j i /" c? 2 v 7 c?v du C d^u i 10. Prove that / u 2 dx=u v + / v-dx. 11. Integrate / (a sin% + 26 sin x cos x + c cos 2 ^)e*^ [a, 1883.] 54 INTEGRAL CALCULUS. 12. Show that if u be a rational integral function of x, where the series within the brackets is necessarily finite. [TRIN. COLL., 1881.] 13. If u I e ax cos bxdx, v I e ax sm bx dx, prove that and that (a 2 + & 2 )0* 2 + v 2 ) = er a *. 14. Prove that - m+1 m+L Also that (m+1) 2 3 (-ir-^! ? ^"- 1 where I stands for log x. 15. Prove that (i.) {e^w J +^""j )&2 le ax ^ n -' 2 bxdx. a?+ri 2 b' 2 J [BERTEAND.] 16. Evaluate / x* log(l - x^dx, and deduce that iT5 + 277 + 3T9 + - == 9""3 10ge2 ' [a, 1889.] CHAPTER V. RATIONAL ALGEBRAIC FRACTIONAL FORMS, PARTIAL FRACTIONS. ALGEBRAIC FRACTIONAL FORMS. 54. Integration of \ and - - - 9 (x or < 4ac. Thus if 6 2 > 4ac, or 7 _ . coth" 1 ; If b 2 < 4>ae, I = / tan " l -.- or -- cot ~ These expressions differ at most by constants, but in any given case a real form should be chosen. RATIONAL ALGEBRAIC FRACTIONAL FORMS. 57 56. Integrals of expressions of the form * px + q can be obtained at once by the following transforma- tion pb px + q _p (2ax+b) , 2a ~~ the integral of the first part being ^ log (ax 2 +bx+ c), Za and that of the second part being obtained by the last article. [The beginner should notice how the above form is obtained. It is essential that the numerator of the Jirst fraction shall be the differential coefficient of the denominator, and that all the #'s of the numerator are thereby exhausted.] T? ' = J log(^ 2 + 4*7 + 5) - 2 tan- 1 ^ + 2). 57. Although the expression px + q may be thrown into the form by inspection, we might proceed thus : Let pa?+gsX(2oaj+6)+/i, where X and /x are constants to be determined. Then by comparing coefficients, giving X = and pb = -- 58 INTEGRAL CALCULUS. EXAMPLES. Integrate 1. f 2 xdx . 4. f fo+ 1 )^ //y> /7/y. /" f/v I \2 ^ a ^ . 5. / Jfl-LZ-^p. x* + 2x+l J x 2 +? 3. / ^t 1 c^ 6. 58. General Fraction with Rational Numerator and Denominator. Expressions of the form A~4, where f(x) and (#) 9w are rational integral algebraic functions of x, can be integrated by resolution into Partial Fractions. The method of putting such an expression into Partial Fractions has been discussed in the Differential Calculus for Beginners, Art. 66. When the numerator is of lower degree than the denominator the result consists of the sum of several such terms as A A Ax+B Ax+B and And when the numerator is of as high or higher degree than the denominator we may divide out until the numerator of the remaining fraction is of lower degree. The terms of the quotient can in that case be integrated at once and the remaining fraction may be put into Partial Fractions as indicated above. A Now any partial fraction of the form - integrates at once into A log (x - a). A Any fraction of the form -. ^ integrates into i x ^~* a) 1 A rl (xa) r ~ v RATIONAL ALGEBRAIC FRACTIONAL FORMS. 59 Any fraction of the form g4 -- has been dis- cussed in Art. 56. ax*+bx+e And when any repeated quadratic factor such as [(x + a) 2 +b 2 ] r occurs in (x) giving rise to partial fractions such as YT~ \2 i 7>2ir we ma y integrate such [_(# + #) ~rfr J a fraction by the substitution x + a = b tan 6, by aid of Art. 67 or Art. 83. But it is frequently better to factorize (x+a) 2 +b 2 into its imaginary conjugate factors x + a-\-ib and x+a tb, and obtain conjugate pairs of partial frac- T) I S~\ T) /~\ tions of the form , - - r^r^.+7 ; -- ^r\i which may (x+a+ib) r r then be integrated and the result reduced to real form by aid of De Moivre's Theorem, as in Art. 63, Diff. Gale, for Beginners. 59. Ex.1. Integrate [ .. * 2+ ^+? - dx. J (x-a)(x-b}(x-c) We have a 2 +pa+q _ _ (x-a)(x-b)(x-c)~(a-b)(a-c)x-a (b-c)(b-a)~^b c*+pc + q . 1 _ y a?+pa + q 1 ^(c-a^c-b^x-c-^a-b^a-c^x-a: 7 ' and the integral is ]g a ' + ggg log(^7 - a). (a-b)(a-c) Ex. 2. Integrate L - ^ - ,dx. J(^-l Let _ Then A(x* + 4) + ^ + "^ 3. Integrate (i) f dx (iii) f W J (^2 +a ^ 2+62 y ' J 4. Integrate (i.) ( xd * . (iii.) [' - v ; J^+^ 2 + l v J^+l do?. (iv.) f A a?2 "t 1 cto. J^ 4 -^ 2 +l RATIONAL ALGEBRAIC FRACTIONAL FORMS. 63 5. Integrate /. v xdx . . x dx (vii.) dx (iiL > (^&T4)- ^ W ^<**> () ( x \ V A */ ~(~& i i\/ (x*+ lX-^-4)' 6. Integrate ~3,J~* J~. (VI.) -7 : x \ d^t? f \ \ / /T ~ t \o/i j ^ j ~t o\' V*-'"^*^ (viii.) '' (#-l) 2 (#2 + iy J ^2 + 1)3 * 7. Evaluate /Vtan~^(9 and P\/c 8. Obtain the value c o cos x dx 9. Investigate 10. Show that r. _f^ fa o 64 INTEGRAL CALCULUS. 11. Prove that 2?r a + b [+* dx _2?r J (x* ax+ 2 X^ 2 bx + b' 2 ) ~~ V3 ab(d> [COLLEGES 7, 1891,] 12. Show that the sum of the infinite series can be expressed in the form and hence prove that [OXFORD, 1887.] 60, CHAPTER VI. SUNDRY STANDARD METHODS, f doc i. Integration of -y= where R = ax 2 +2bx+c. Case I. a Positive. When a is positive we may write this integral as If dx a a which we may arrange as dx If dx If I __ ____.^ = ._. = _..___^_^ == i Q p _ __ I . aJ 7/ , &\ 2 b z -ac x/J +- according as 6 2 is greater or less than ac, and the real form of the integral is therefore (Art. 36) , ., ax + b 1 . , T ax + b = cosh ~ * * or 7^ smh ~ 1 , , ^ Vo 2 ac Va x/ac 6 2 according as & 2 is > or < E. T. C. 66 INTEGRAL CALCULUS. In either case the integral may be written in the logarithmic form ~T= log (ax + b + *Ja*Jax 2 + 2bx+c), ^ _ the constant T= log v 6 2 ~ ac being omitted, *J a Also since cosh ~ l z = sinh ~ l \/z 2 1 , and sinh ~ l z = cosh ~ l \/z 2 + 1 , 1 , , ax + b 1 . , \/aR = cosh ~ l . = =. sinh " x I . , , ax + b 1 , T x/aJi and -7=. sinh " - 1 7-- = 7^ cosh 1 - 7 - ? V^ \/ac b 2 \/a \/ac b 2 which forms therefore may be taken when a is positive and b 2 is greater or less than ac respectively, 61. Case II. a Negative. f dx If in the integral / ' . a be negative )*Jax 2 +2bx+c write a= A. Then our integral may be written 1 r dx ZJ or or 7=: sin SUNDRY STANDARD METHODS. 67 or omitting a constant , ax + b n, - 1 -- - -- for -^ 2J' Also since cos ~ l z = sin ~ 1 x/l 2 , we have cos~ \/b* ac V& 2 s that when R = ax z + $ ! / 7? negative, ac. It thus appears that when ^R = a positive. /ac^P' and the real form is to be chosen in each case. Ex. 1. Integrate I , -7- ^ We may write this cfo? . , i# = - smh- 1 ._, v/2 \/23 p=. v/UJSIl . ' \/2 \/23 i.e. the integral = -i= log(4# + 3 + 2 V2 V2^ 2 + 3# + 4) V2 ing the constant - log ). 68 INTEGRAL CALCULUS. dx Ex. 2. Integrate ( J This integral may be written I dx and therefore is = sin" 1 -^^ . \/2 \/41 which may also be expressed as -^cos - F => V2 */41 EXAMPLES. dx 1. Integrate { --^ { JV^ + 2a? + 3 J 2. Integrate /" dx dx J -, A/2- 3.* -2# 2 3. Integrate \ >Ja + Zbx+cx*dx (c positive). 4. Integrate /\/a + 2&# cyPdx (c positive). -. 62. Functions of the Form -===== may x/a^ 2 +26^+c be integrated by first putting Ax+B into the iorm which, may be done as in Art. 57, either by inspection or by equating coefficients ; we obtain Ax+B ex/ SUNDRY STANDARD METHODS. 69 The integral of the first fraction is A and that of the second has been discussed in Articles 60, 61. EXAMPLES. Integrate - 2.37 + 3 x+b POWERS AND PRODUCTS OF SINES AND COSINES. 63. Sine or Cosine with Positive Odd Integral Index. Any odd positive power of a sine or cosine can be integrated immediately thus : To integrate I sin 2n+1 # dx, let cos x = c, .'. smxdx= dc, Hence dc fsin^+^cfo = - ((I-c 2 ) __ 70 INTEGRAL CALCULUS. Similarly, putting since = s, and therefore cosxdx=ds, we have I cos* n+l x dx = (1 s 2 ) n ds nn L_ > ' I / 1 \7l. 64. Product of form sin^ cos?#, p or q odd. Similarly, any product of the form admits of immediate integration by the same method whenever either p or q is a positive odd integer, what- ever the other be. For .example, to integrate / sin 5 # cos 4 # dx, put cos#=c, and therefore - sin xdx= dc. Hence / cos% sin 5 ^? dx / c 4 (l c 2 ) 2 dc cos 5 ^7 9 cos 7 ^; cos 9 ^? ~5~' J ^f~ "T"' /^ sin 5 ^ cos 3 # dx we proceed thus : = I sin^(l - sin 2 x)d (sin x) 65. When p+^ is a negative even integer, the expression sin*to cos% admits of immediate integration in terms of tan x or cot x. For put tan x = t, and therefore sec 2 ^ dx = dt } and let p + q = 2n, n being integral. Thus = | p + i ) n ~ l dt ^ 8 a5 , Irf tan^ +6 a5 4- n ~*-(j __ I- 4-- 2 SUNDRY STANDARD METHODS. 71 Similarly, if we put cot x = c, then cosec 2 ^ dx dc, and \&DPxco&xdx= - a result the same as the former arranged in the op- posite order. Ex. 1. Integrate f?^5 />2n-fl 2w+2 aj dx = - c - c - W C- . . . ~ where c = cot x. 70. Odd positive integral powers of a secant or cosecant can be integrated thus : By differentiation we have at once d and (n + l)cosec n+2 & n cosec 7l o? = -7-(cot x cosec n a doc whence (n + 1 ) sec w+2 ^j dx = tan x sec n # + ^ I se and (ti + 1 ) cose.c n+2 xdx =~coix cosec n x + n cosec n # dx I f / \ Thus as sec x dx = log tanf^ + ^ V and I cosec # c?x = log tan^ , we may infer at once the integrals of sec 3 #, sec 5 #, sec 7 cc, . . . ; cosec 3 aj, cosec 5 #, etc., by successively putting n = I, 3, 5, etc., in the above formulae. Thus / sec 3 # dx = J tan x sec x + ^ log tan f - + ^ V / sec 5 ^? da7= J tan # sec 3 ^+ 1 / sec 3 ^ = J tan x sec 3 #+f tan ^7 sec x + f log tanf -+- ), etc. SUNDRY STANDARD METHODS. 77 71. Such formulae as A are called " REDUCTION " formulae, and the student will meet with many others in Chapter VII. We postpone till that chapter the consideration of the integration of such an expression as sin^cos^ except for such cases as have been already considered. 72. Since a positive power of a secant or cosecant is a negative power of a cosine or sine, and a positive power of a cosine or sine is a negative power of a secant or cosecant it will appear that we are now able to integrate any integral positive or negative power of a sine, cosine) secant, or cosecant. INTEGRAL POWER OF TANGENT OR COTANGENT. 73. Any integral power of a tangent or cotangent may be readily integrated. For tan n x dx = tan n ~ 2 x(sec?x l)dx = Itan n - 2 a3c?tan# ltan w - idM. n ~ l x f, , -^ tan ?l - 2 #cfe. n 1 J And since I tan # cfc == log sec #, and I tan%c dx = (sec 2 # l)dx = tan x x, we may integrate tan 3 #, tan 4 ^, tan 5 #, etc. Thus we have / tan 3 #cfo?= / tan x(sec 2 x- l)dx 78 INTEGRAL CALCULUS. [ f 2 3 By continuing this process we shall evidently obtain _ 2?i-l 2^-3 + (-l) w ~ 1 tan # + (-l) n ff, and tan-+^=^^ _ tan^^ t Similarly I coi n x dx = cot n ~ 2 cot 71 - 1 ^ f = -- r- ~ |COt n - 2 #CfcE, 711 J whilst icot^^aj = log sin x. and cot 2 ^ ^ = I (cosec 2 ^ 1 )dx = cot x # ; and therefore we may thus integrate cot 3 # 3 cot 4 #, cot 5 ^, etc. Hence any integral power of a tangent or cotangent admits of immediate integration. f dx 74. Integration of \ a+bcosx , etc. We may write a + b cos x as s 2 | - sin 2 | j, SUNDRY STANDARD METHODS. 79 or Thus or (a (fa 2 = -AgU-j 6 (1) CASE I, If a > b this becomes tan i a- 6 /a+6 or Since we may write this as 1 , tan ? | 2J- 1- b, =5- COS ^ g + b 2 80 or INTEGRAL CALCULUS. 1 COS" 7 *- + bcosx CASE II. If a < b, writing the integral in the form (K , dian.~ (2) in place of the form (1) we have in this case by Art. 54 UjiAj J. J ft _1_ A f*na sv* A n IT. 1 17 i + bcosx b a Ib + a Ib + a x Vf^ V6^~ tan 2 , v6 + a + \/b a tan = > lo g . . ' rjr> \J~b + a v b a tan ^ By Art. 33 this may be written tanh~ : or, since /6 2 -a 2 2 tanh ~ l z = cosh ~ 1 1 2 ' we may still further exhibit the result as b a. nX 1 -L-^ tan 2 ^ b + a 2 :cosh~ 3 1 1 ^- b + a or SUNDRY STANDARD METHODS. We therefore have b, x 81 dx a+ bcosx i.e. or Jl?- = cosh- a+bcosx er < b. These forms are all equivalent, but one of the real forms is to be chosen when the formula is used. 75. The integral of -r may be im- a+ b cos x + c sin x mediately deduced, for b cosas-fc smx = \/b 2 -{-c 2 cos(x tan~V ), \ b/ and therefore the proper form of the integral can at once be written down in each of the cases a greater or less than ^/5*+c*. dx [ dx Ex. = f in x J 13 4- 3 cos x H- 4 sin x 1 /13 2 - 5 2 13 + 5 coe(# - a) -- (where tana = ^) :-a) _ 12 or 1 -i/2 .'T a\ $. I. C. 82 INTEGRAL CALCULUS. f dx 76. The integral I , 7 . -- may be easily deduced Ja + 6sm# by putting f dx f dv then j. =B I- ^- - , J a + o sin x J a + o cos y and therefore its value may be written down in both the cases a^b. Of course it may be investigated also independently by first writing a + b sin x as a(cos 2 | + sin 2 |j + 26 sin | cos |, or cos 2 ^( a + 26 tan - + a tan 2 ^ J. The integral then becomes 2 and two cases arise as before. 77, The integral I , x , may be similarly treated. dx f dx SUNDRY STANDARD METHODS, 83 2 lb a,,x ifb> a ) this = - which further reduces to ,6 + a cosh # 3 *~ A___ * a + fccosha' and if b < a the integral is 2 , , la b, , ^ 7-27-2 tanh " V r, 4. /) tanh 9' 'a 2 6 2 ^a+o z which further reduces to 7===- cosh ~ a + b cosh x 78. Similarly the integrals of 1 and of a + bsinhx a + bcoshx + csmhx may be easily obtained. EXAMPLES. , T , ( Vtan# 7 1. Integrate / ax. J sm#cos# 2. Integrate (i.) f 8ecxdx . J a + b tail x (ii.) I -. ; = r- 9 . J (a sin x + o cos xy 1 3. Integrate (i.) 84 INTEGRAL CALCULUS. 4. Prove that, with certain limitations on the values of the constants involved /"%= =L= - J J(a-x)(x-(3) P .-(3 ____ and integrate / v (x a)(/3 x)dx. a 5. Integrate ,. v C dx , r \ f >} J SC &^% ' (1 " } 3(l-s (') . .r . ( v -) - 2^2 4- cos a? + sin .77 / \ r Ou27 / * \ r ^^ U11 '-' J cos a + COS.T' ^'''^ J 2 siii 2 <9 + 6 2 cos 2 ^' (vii.) cos a + cos x and (viii-) prove 6. Integrate (i.) f- o dk C?Jt' (ii.) / ____ ^ V a(^ - b) + V 6(^ - a) (iii.) f <** J 7. Integrate f 7 I 8. Integrate f- - ^ - . J sm^ + sm2^ 9. Integrate fcos201og cos ^ +shl fa. J COS0-S1TL0 10. Interate 1+cosx SUNDRY STANDARD METHODS. 85 11. Integrate 12. Integrate / sm x _dx. J VI + sin x 13, Integrate / sec ^ dx. J 1 + cosec x 14. Integrate /- f^fl_f_. J v a + b tan 2 # 15. Evaluate fVr^ ^' / 1 + sin x o 16. Integrate [****"****&,. J log tan ^7 17. Integrate . Vsin 2(9 18. Integrate fcot0-3cot30 J 19. Integrate / ~ J Wo? 20. Integrate /" 7 J (x si 21. Integrate f-^ 22. Integrate f A /_ * ~ CQS ^ 6 y ' V cos (9(1 + cos 6>)(2 + cos 6>) f 23. Integrate 24. Integrate f- ^ (sin + cos 25. Integrate f J . . 1 sin ^? " 2 sin a? mg-coeg INTEGRAL CALCULUS. 26. . Integrate I sin" 1 - dx. J l+x 2 27. Integrate ? J \ 28. Integrate ( sin ^, [*- X -dx, {^^dx, and prove that J sin 2# ' J sin 3^ J sin 4^7 sin&r 5 + [THIN. COLL., 1892,] CHAPTER VII. REDUCTION FORMULAE. REDUCTION FORMULAE. 79. Many functions occur whose integrals are not immediately reducible to one or other of the standard forms, and whose integrals are not directly obtainable. In some cases, however, sucft integrals may be linearly connected by some algebraic formula with the integral of another expression, which itself may be either im- mediately integrable or at any rate easier to integrate than the original function. For instance it will be shown that (a 2 + # 2 )^fe can be expressed in terms of J(a 2 + # 2 )^fe, and this latter itself in terms of J(a 2 + ar)^cfe, which being a standard form the integral of I(a 2 + # 2 )^cfc may be inferred. Such connecting algebraical relations are called Reduction Formulae. 80. The student will realise that several reduction methods have already been used. For instance the 88 INTEGRAL CALCULUS. method of Integration by parts of Chapter IV., and the formulae A of Art. 70. It is proposed to consider such formulae more fully in the present chapter, and to give a ready method for the reproduction of some of the more important, 81. On the integration of x m - l X* where X stands for anything of the form a+bx n . In several cases the integration can be performed directly. I. If p be a positive integer, the binomial in expands into a finite series, and each term is integrable. /v Next suppose p fractional = -, r and 8 being integers and s positive. 777/ II. Consider the case when is a positive integer. Let X = a + bx n = z s , .'. bnx n ~ l dx = sz s ~ l dz z s ~ l f \x J and r - bn) and when is a positive integer, this expression is directly integrable by expanding the binomial and integrating each term. III. When is a negative integer, the expression (z s -a)~ n+ ' REDUCTION FORMULAE. 89 may be put into partial fractions, and the integration may then be proceeded with (Art. 58). TD r IV. If H is an integer positive or negative, we may proceed thus : , rn _ - rn m-\ -- and by cases II. and III. this is integrable when - is either a positive or a negative integer by the substitution b + ax~ n =-z s . That is, the expression is 777 r integrable when \-- is integral, positive, or negative. n S Three cases therefore admit of integration im- mediately or by simple substitution. (1) p a positive integer. (2) an integer. 777 (3) [-p an integer. Ex. 1. Integrate (^(c Here m=6, n = 3, and =an integer. n Let so that %x*dx= 2zdz. Then the integral becomes 90 INTEGRAL CALCULUS. Ex. 2. Integrate / x*(a? + x^dx. Here m = , n = 3, p=b and +p is an integerc n ' The integral is / Let then -3-. XT and the integral becomes 9 * which might be put into partial fractions. If, however, z be put = sec 6, the process of putting the expression into partial frac- tions will be avoided and the final integration may be quickly effected (Art. 70). 82. Reduction formulae for \x m ~\a Leta + 6x H = ^C; then \x m ~ l X^dx can be connected with any of the following six integrals : x m - \X P - 1^ \ x m-n- IXPdx, x m - n - lX? +l dx, X m+n ~ 1 according to the following rule : m+n - Let P = X+1 JT ya+1 where A and JUL are the smaller indices of x and X respectively in the two expressions dP whose integrals are to be connected. Find -p. Re- arrange this as a linear function of the expressions whose integrals are to be connected. Integrate, and the connection is complete. REDUCTION FORMULAE, 91 Ex. 1. Connect tx m - l X\lx with (x^ Let P=x m X p . Then = dx [Note the rearrangement "as a linear function, etc., etc."] Hence P=(m +pn) ftf- l A'*dx - apn I x m -^X p ~ l dx, or [^X'dx-****- + -22?- (ar-*X*-*dx. J m+pn m+pnj The advantage of this reduction is that the index of the usually troublesome factor X p is lowered ; and by successive applications of the same formula we may ultimately reduce the integral to one which has been previously worked, or which can be easily obtained. Ex. 2. Thus, for instance, to find l(& + cFydx we may con- nect this integral with / (x 2 + cfifdx, and this again with / (# 2 + aPfdX) and this last is a standard form. As the reduction is used twice, we will connect ((a?+arfdx with (( Let P=x [Note the preparatory step which might be performed mentally \ = (n + I)(x 2 + arf -na\x* + arf ~ l [which is now "rearranged as a linear function, etc., etc."\ 92 INTEGRAL CALCULUS. Integrating, P=(n + 1) I(x 2 + a?fdx-na? /"( and n+I Putting ?i = 5 and ft = 3, (( J and Then 3i 6.4 Ex. 3. Calculate the value of [^ x^-^ax-x^dx, m being a positive integer. We shall endeavour to connect l% m >JZax-x'*dx with f x m ~ l *J%ax-x*dx y i.e. (x mJf ^(Za-x?dx with ( x m ~\2a-x*fdx. Let P=^ m+1? (2a-^) 1r according to the rule, then Hence (m + 2) fx m+ ^(2a - xfdx - x m ^(2a - ^ + (2m + l)a fx m ~\2a - xfdx o REDUCTION FORMULAE. 93 a . x m *J?Ltix ra + 2 Jo m + 2 o /la, . _ x m *j%ax - x*dX) and m be a positive integer, 2ra-l 2 2m-l 2m -3 3/ _ . . - -- . - a J. m -z etc. m + 1 m 2m -1 2m-3 5 3 m "mT2 ' m + 1 ' m "*4 ' 3 ' Now to find I Q or I fj^ax x^dx, put x=a(\ cos 0). Then dx = a sin ( an d ^l^ax x^ a sin 0. Also when #=0, we have $=0, when # = 2a, we have O = TT. Hence 7 = fVsinW^- T(l - cos 20)rf0 Hence / -m- 1)...3 +2? r_ (2m + l)! (m+2)(m + l)...3 2 m!(m + 2)! EXAMPLES, Apply the rule stated in Art. 82 to obtain the following reduction formulae (when X=a + bx n ) : 1. / J 2. 94 INTEGRAL CALCULUS 3. (,-^ J 4. ( J . (a*-*X*dx = xm ^ - ^P {x+ n J mm] . / 6. Integrate out 7. Obtain the integrals of / x m ^(^Lax x^dx for the cases m = l, m=2, m = 3, and their numerical values when the limits of integration are and 2a. 83. Reduction formulae for sin^a? co&x dx. A similar rule may be given for a reduction formula for I siu p x cos q x dx, j This expression may be connected with any of the following six integrals : I sin^ " 2 # cos?# dx, \ si \ sin p x cos 9 ' ~ ^x dx, \ si I sin^ - 2 x cosv +2 x dx, sin^+ 2 ^ cos^ - *x dx, by the following rule. Put P = sin X+1 #cos AA+1 & where X and ^ are the smaller indices of since and cos# respectively in the two expressions whose integrals are to be connected. Find -T-, and rearrange as a linear function of the ax expressions whose integrals are to be connected. REDUCTION FORMULAE. 95 Integrate and the connection is effected. Ex. Connect the integrals / Let P=s =(p l)sm p ~ 2 x cos g ^(l sin 2 #) (q -f l)si = ( p I )sin^~ 2 # cos 9 # ( p -h ^)sin^ cosPx [Note the last two lines of rearrangement as a linear function of sin^cos^ and sin^~ 2 ^7 cos%], . * . P= ( p - I ) / siii^~ 2 ^7 cos 9 ^ dx - ( p + q) I si Hence / sin.^ cos%^ = - 8m *~^ X cosq+l * + zi ( * p + q P + qJ It will be remembered, however, that in the case where either p or q is an odd integer the complete integration can be effected immediately [Arts. 64, 67]. The present method is useful in the case where p and q are both even integers. EXAMPLES. Connect the integral fsin*4?cos?#ei? with 1 . / sm p+2 x cos q .v dx. 2. / siii^ cos~ 2 3? dx. 3. I sm p x 4. /*sin^- 5. / sin^ + 2 # cos~ 2 # dx. 96 INTEGRAL CALCULUS. 6. Prove that fsin^^ _ cousin- 1 * -I f^,. J n n J Employ this formula to integrate sin%, sin 6 #, sin 8 ^. 7. Establish a formula of reduction for / cos w # dx, 8. Integrate sin 4 84. To calculate the integrals V 71 f 2" . fl 5^ n = I sm n #? aa? and (7 n = I J J Connect I sin n ^c dx with I sin n ~ 2 x dx. Let P = sin n ~ 3 #cos;E according to the rule; then dP su n x dx = (n l )sin n ~ 2 x f . sm n ~ l xcosx , n If . r .*. lsin n ^a^= --- -\sm n ~ z xdx. J n n J Hence since sin -^ cos & vanishes when ?& is an integer not less than 2, when x = 0, and also when x = J, we have 711 ^ 3 71 5 a = --- ^ --- ^ * >w- 71 712 7i 4 if 71 be even this ultimately comes to Ti-l Ti-3 5 3 Iff "~'ii^V"g 4 2j J that is REDUCTION FORMULAE. 97 Ti-1 7i-3 3 1 TT n n -2 '" 422' If n be odd we similarly get 1 Q A, 9 / 2" f-j il/ ^^ -L ll/ ^^ O TJ Z< I . -| .v . I dm w rl w Ufl 6e ei>e7i = 2m, and # afeo even = 2n, (2m-l)(2m-3) /( ) = 2r v 3)...l m 2 v p). CASE IV. Ifpbeodd =2m l,and g odd =2n 1, (2m-2)(2m-4) ^ 2 ^_i ) = e tc. 27i-2)(2m + 27i-4/ v (2m-2)(2m-4)...2 ( } 2^-2)(2m + 2w-4)...(2^ + 2) A 7T and /(I, 2w - 1) = Tsin (9 cos 2 "-^ dO = f - 5^-f] . I-, J L 2w Jo 2 86. Expression in a single rule. These four formulae may be expressed under one rule as follows : Let r(n + l) be a function defined by the relations 100 INTEGRAL CALCULUS. These relations will be found to sufficiently define T(n + 1) where n-{- 1 is either an integer or of the form 2k + 1 2 ' k being a positive integer. For instance, T(6) =5F(5)= 5 . 4F(4) = 5 . 4 . 3r(3) = 5 .4.3. 2F(2) = 5.4.3.2.ir(l) = 5! V-) =F(f )= S . PXiHf . I- . f r(f )= i . | . f . f r( ) This function is called a Gamma function, but we do not propose to enter into its properties further here. The products 1.3.5... 2n-I 2.4.6 ... 2u TT which occur in the foregoing cases of I sin^0 cos?0 d9 o may be expressed at once in terms of this function. ^ ^(2n+l\_2n-l 2n-3 2n-5 l r /l\ 1 \~~2~) 2 2 2~ ' 2 V2/' so that and sothat /y 7T 2 Hence in Case I. REDUCTION FORMULAE. 101 In Case II. In Case III. we evidently have the same result. In Case IV. It will be noticed therefore that in every case we have the same result, viz., 7T f and that the ^ +1 occurring in the denominator is ?9 + l #4-1 the sum of the 1 and the ^ in the numerator. This is a very convenient formula for evaluating quickly integrals of the above form. IT Thus f \in6 cos 8 dO = -* r s ., __f f ' j- V^TT j" f * f i| ^/?T _ 5?T 2.7 - .6.5.4.3.2.f~~2 15 ' 102 INTEGRAL CALCULUS. 87. The student should, however, observe (as it has been pointed out previously), that when either p or q or both of them are odd integers, the expression sinP$cos?# is directly integrable without a reduction formula at all. For instance, (sin 6 6>cosW6> = [sin^(l-sin 2 6'Xsm6'=^ 761 -^^, J J 79 and Similarly, cos 2 6>(l -2 cos 2 <9+cos 4 6>)dcos "i COS 3 <9, Q COS 5 <9 COS 7 6n .jo,! s - 3 +2 Jr 4 -"-*^*- But when p and q are both even and the indefinite integral required, or if the limits of integration be other than and ^, we must either use the reduction tt formula of Art. 83 or proceed as in Art. 67. EXAMPLES. Write down the values of 1. f- /> ***** , / sin 8 o;^ *, / COS 9 ^70 2. C- iiii 6 j? cos x dx, I sir i*co#.r cos- n c/> + M - ~ \ f cos 2 / ^^t \ 2iii / J (b) 104 INTEGRAL CALCULUS. 2. Investigate a formula of reduction applicable to when m and n are positive integers, and complete the in- tegration if ra=5, 7i = 7. [ST. JOHN'S COLL., GAME., 1881.] 3. Investigate a formula of reduction for and by means of this integral show that ._J_ a dinf 271+2 2 27i + 4 2.4 271 + 6 2.4.6 271 + 8 2. 4. 6. ..27i ~~3. 5.7...27i + l' Sum also the series 1 1 1 1.3 1 1.3.5 1 , . \f I m , I t I > _1_ OjCll ITlrT 271+1 2 2^ + 3 2.4 27i + 5 2.4.6 271 + 7 [MATH. TRIPOS, 1879.] 4. Prove that 2n+l / (rf *-, prove 6. Find reduction formulae for (a) x(a + bx) P *dx, (y) J /2p+l *^*+a)*r, ( S ) and obtain the value of /*(* - . [COLLEGES> CAMR] 7. Find a reduction formula for I e ax cos n x dx, where n is a positive integer, and evaluate [OXFORD, 1889.] RED UCTION FORMULAE, 1 05 8. Find formulae of reduction for / # w sin x dx and / e ax sin n x dx. Deduce from the latter a formula of reduction for Jcos a* 8in"* " f 2 n+1 In n( 1) (^ l)(n-2) (2ro-lX2ft-3)...3 TT ' 8' [MATH. TRIPOS, 1878.] 10. Show that 1 V 1 \ / ' wi 2/3 n2fifJ) [TEIN. COLL., CAMB., 1889.] 11. Prove that 7 _1.3.5 ...(2w-l) TT 2.4.6 ...2m 1 1 * ~2.4.6...2m 4~3.5.7...(2m + l) ' 2* 12. Find a formula of reduction for f-~=L Show that ^ v'^ 1 3. 5. 7. ..(27i+l)l where a 1} a 2 , ... are the binomial coefficients. [ST. JOHN'S, 1886.] 13. Show that 2 TO / cos mx cos m # dx / = r+- 4- sn^ mm sn^ , ~~ ~T72 ~" where m is an integer. [COLLEGES a, 1885.] 106 INTEGRAL CALCULUS. 14. Show that m being a positive integer. [OXFORD, 1889.] 15. Prove that if Im, n = I cos m # sin nx da:, (m + n)I m) n = cos m x cos nx + m/ m _i ? n _i ? [if 1 / 9 2 93 9i\ ^-l^( 2 +i+i + - + ^J- [BERTRAM] 16. If / m? w = I cos w # cos nx dx, ,1 , r cos%^7 d f cos m x\ m(ml) T prove that / m w = - ^ - ^ I - - ) + v ./^m-2, n, m 2 -^ 2 cEr\oos9u;/ m*-n 2 and show that /I cos m ^7 sin 7^^; dx\ prove that ^ m w = -- 1 w m _i M _I. ' m + ?i m + n Hence find the value (when m is a positive integer) of /If cos m # sin 2mx dx. [7, 1887.] r 2 IT 18. Prove that / cos n x cos nx dx= T J 2 n+1 [BKRTRAND.] 19. If m + n be even, prove that / co o ' - m-n. i . - 1 2 2 [COLLEGES, 1882.] REDUCTION FORMULAE. 107 20. Evaluate the integral / c [COLLEGES, 1886.] TT V TT 21. If / cos m # cos nx dx be denoted by/(m, 71), show that [OXFORD, 1890.] 22. Prove that if n be a positive integer greater than unity, ft I I cos n ~ 2 x sin nx dx = - o /"sin nx-. Ti-l* [OXFORD, 1889.] 23. Find a reduction formula for the integral / ^ in - J smo? 24. If u m) n = I - fLJlfl&pj where m is not less than ft, and TTC, 7^ are either both odd or both even integers, show that (n l)(n 2)u m) n + m?u m) n _ 2 - m(m - l)u m - 2) n - -2 = 0. 25. If u n = show that u n = , - ._: + J5 M n - where A= - -3) a ^ 7i-2 1 = - - - =-, = , = - - - n 1 a 2 b 2 (n l)(a 2 o 2 ) n I Show that I* - J (l-e 2 s (i_ e 2)| 16 e being less than unity. [ST. JOHN'S COLL., 1885.] 26. Prove that f sinW and dashes denoting differ- entiation and suffixes integrations with respect to x : / / u + (- l) n - 1 nu^ n - 1 h n+ i + (- l)"n I uMv n+ idx + (- l) n {dx (u^v n dx. [a, 1888.] CHAPTER VIII. MISCELLANEOUS METHODS AND EXAMPLES. f dx INTEGRALS OF FORM \^ . 88. The integration of expressions of the form dx can be readily effected in all cases for which I. X and Y are both linear functions of x. II. X linear, Y quadratic. III. X quadratic,- Y linear. If X and Y be both quadratic the integration can be performed, but the process is more troublesome. 89. CASE I. X and Y both linear. The best substitution is : Let -fe dx 110 INTEGRAL CALCULUS. Putting , cdx , we nave ,_ = at/. and ax + b = -(y 2 e) + b, C and / becomes 21 ^ . 7 , which, being one of jay 2 ae + bc the standard forms 2 _^ 2 , is immediately integrable. Jy A Ex. Integrate /= f J (x- Let then Thus y-l y+lj 90. The same substitution, viz., *Jy=y will suffice l(fi( T\f] 'IT for the integration of I -^ ^ when ^(cc) is any rational integral algebraic function of x, and X and F are eacfe linear. . Integrate /== f J^- Ex. Writing >/^ + 2=7/, we have %dy and .r = ?/ 2 - 2, MISCELLANEOUS METHODS AND EXAMPLES. Ill so that -L ==& + 24/-32/ + 16 (by common division). Thus 91. CASE II. X linear, F quadratic. The proper substitution is : Put X=\ y Let Putting ax + b = - , t/ we have, by logarithmic differentiation, adx dy ax + b y and ex 2 + ex +/= - 2 (- - 6 Y + - (- - ft) +/ a 2 \/ / a\ / Hence the integral has been reduced to the known form /= : which has been already discussed. 112 INTEGRAL CALCULUS. Ex. Integrate /= f ./ Let #+l=y-i, then __=_ an d #+1 y /= __ i+i-2 i+%-y 2 JI* V 92, It will now appear that any expression of the form f J( can be integrated, J (ax 2 + bx + c)\/ex +f where 2 \ /j/2nj_ \ / , / 2n-2_i _i_ >L the form "^ - ' which by divisi n ' and the rules for partial fractions, may be expressed as and each term is at once integrable. Ex. Integrate /= Putting \/^ + l=y, we have - 7 ====2e?y, and v^ + 1 _ 2 MISCELLANEOUS METHODS AND EXAMPLES. 115 EXAMPLES. Integrate the following expressions : 95. CASE IV. X and F both quadratic. We do not propose to discuss in general terms the method of integration of expressions of the form f ^) dx a where X and F are both quadratic and (x) rational, integral and algebraic, as it is beyond the scope of the present volume. We may say, however, that the proper substitution for such cases is A/y : = ^' anc ^ ^ e student will glean the method to be adopted from the following examples.* Ex. 1. Integrate 1= f Putting y ~dx dy (a 2 * The student may refer to Greenhill's "Chapter on the Integral Calculus" for a general discussion of the method. 116 INTEGRAL CALCULUS. Thus /becomes (a 2 W Also so that and Thus / reduces further to If a > 6, we may arrange / as i / V^TP , - Ex. . 2. Integrate /= / - J 2x 2 -2^ __ J (2x 2 -2^+1) \/3^ 2 -2^ + 1 1 dy _ 3^-1 2# 1 ~~ -, , . The maximum and minimum values yj 2 and yf of ;/ 2 are given by x = \ and # = 0, and are respectively 2 and 1, so that for real values of #, ?/ 2 must be not greater than 2 and not less than 1. MISCELLANEOUS METHODS AND EXAMPLES. 117 Now yi-f=^-y"=' 2^ 2 and t-tfm^l-g Thus / becomes _ r(&g 3 - 2ar+l )(2a7 2 J x(x-l) /2# - 2.37+1 Now ~ 1 Thus '=/(-,== 2 = cosh" 1 ?/ + 2 cos" 1 -^- , 1 /3^ 2 - N /2 \2^- Integrate 1. 2. 3. EXAMPLES. 4. _4 5. 6. 118 INTEGRAL CALCULUS. 96. Fractions of form CT +f ina; + CCOSa; . a l + Pi sin ^ + CfiOSX This fraction can be thrown into the form A 1 1 ~ (Oj + ^sin x + c 1 cos x) (a x + b^siu x + c-fos x) where A, B, G are constants so chosen that A + Ca^a, -Bc^ + Cb^b, and each term is then integrable. 97. Similarly the expression a + b sin x-\- c cos x may be arranged as (a +6^+0 cos .)"+ _| - (Oj + 6 x sin x + c-[Cos x) n ~ and the first and third fractions may be reduced by a reduction formula [Ex. 25, Ch. VII.], while the second is immediately integrable. 98. Similar remarks apply to fractions of the form a + b sinh x + c cosh x a + b sinh x + c cosh x a i + frisinh x + Cjcosh #' (a x + 6 1 sinh # + qcosh x) n ' 99. Some Special Forms. It is easy to show that sin a? Isin^r c\ sin a 'sin(a 6)sin(a c) . a), MISCELLANEOUS METHODS AND EXAMPLES. 119 sin 2 ,^ and 7 r -. -. T\. -, r a)sin(^ 6)sm(# c) sin 2 a 1 ^^i 1 1/ , / __ r)\nY\( rt /^ QTTn / 7* /Y i E51111 Iv ^^ U lollll (-t/ ^^ I/ I ollll cC< ^^ tv^ f sin # c?a? whence I -r, . M / . \ Ssina . . -^7 7 . . , r lo^ sm(o3 a), sm(a 6)sm(a c) 6 v , - 7 v . / - r (a 6)sm(a c) , f sin 2 ^ dx and -5-7 -- r-^-p - , x . / J sin(^ a)sm(x o)sm(x S - ; - 1\ / sin (a 6)sm(a r c) sin 2 a x a tan ; - 1\ / - : 7r . (a 6)sm(a c) 2 100. More generally Hermite has shown * how to integrate any expression of the form _ /(sin 8, cos 9) _ sin($ a 1 )sin(0 a 2 ) . . . sin(0 a n )' where f(x, y) is any homogeneous function of #, y of Ti 1 dimensions. For by the ordinary rules of partial fractions f(t, 1) _ /(a,, 1) _ (* - Oj)(t - a 2 ) . . . (t - a^ (a x - egCoj - a 3 ) . . . (a x - ) x , ^ a x (a 2 ^Xag a 3 ) ... (a 2 a n ) ^ a 2 which may be written __ _ ^r((a r a 1 )(a r a 2 ) ... (a r a w ) ^ a r (the factor a r a r being omitted in the denominator of the above coefficient). * Proc. Lond. Math. Soc., 1872. 120 INTEGRAL CALCULUS. Putting = tan$, a 1 = tana 1 , a 2 = tana 2 , etc., this theorem becomes /(sin 0, cos 6) sin($ a 1 )sin(0 a 2 ) . . . sin(0 a n ) /(sin a r , cos a r ) r= isin(a r OL) ... sin(a r a n ) sin($ a r ) Thus W: /(sin 0, cos 9) /(sin or, cos a,) _ logtan ^ " x ./ ~ 7 ^ 7 \ J-'-'ii t/ctj-j. ;-: . ^ism(a r %) . . . sm(a r a n ) 2 EXAMPLES. Integrate sm ^ 4 cos ^^7 - cos x cos a cos 2# cos 2a K sin 2^7 sin 2a cos ^ cos a sn ^7 sn a cos 3# cos 3a O. . D. cos x cos a sin #(sin 2 ,# sin 2 a)' GENERAL PROPOSITIONS. 101. There are certain general propositions on integration which are almost self evident from the definition of integration or from the geometrical meaning. Thus f (j)(x)dx= J 102. I for each is equal to \^(^) ~~ \H C if ( x ) he the differ- ential coefficient of \fs(x). The result being ultimately * See Hobson's Trigonometry, page 111. MISCELLANEOUS METHODS AND EXAMPLES. 121 independent of x it is plainly immaterial whether x or z is used in the process of obtaining the indefinite integral. /& pc /& 103. II. 1 anc ^ ^ e ^ ^ ne or " dinates N^^ NJP^ N^P^ be cc = a, x = c, x = b respect- ively. Then the above equation expresses the obvious fact that Area + area 104. III. [ (x)dx = f 0(a - x)dx For if we put x = a y, we have dx = dy, and if x = a, y = 0, Hence I (a y)dy o a = fV-2/X2/ (by in.) o = I <{>(a x)dx (by I). o Geometrically this expresses the obvious fact that, in estimating the area 00' QP between the y and x O' Fig. 9. axes, an ordinate O'Q, and a curve PQ, we may if we like take our origin at 0', O'Q as our F-axis, and O'X as our positive direction of the X-axis. MISCELLANEOUS METHODS AND EXAMPLES. 123 p2a pa pa 106. V. (x)dx=\ (f)(x)dx+\ (j)(2a x)dx. 00 For by II. rpa p2 (x}dx + \ and if we put 2a x = y, we have dx = <%, and when x = a, y = a, when x = 2a, y = Q. /2tt pO 0(a)c/x = - 0(2a - y)dy a a Thus f a = I (2a x)dx. o fa pa pa 0(#)cfo=l 0(^)cZa;+ (2a x)dx. ooo We leave the obvious geometrical interpretation to the student. 107. VI. Plainly if 0(a?) be such that this proposition becomes p2a pa I (x)dx = 2\ (2a a?)= ti /net pa \ (x), it is clear that it consists of an infinite series of repetitions of the part lying between the ordinates OP (x = 0) and JV^Pj (x = a} and the areas bounded by the successive portions of the curve, the corresponding ordinates and the #-axis are all equal. Thus f <{>(x)dx= r'(t>(x)dx= f jwa /a I ^>(x)aa; = 71 1 , j A T - n 7 A%n-I 2n-3 I IT sin xdx=% \ sm \276u? =4 / Bin #aa?*4 ...- -. J J J 2?^ 2ra- 2 2 2 O Fig. 10. SOME ELEMENTARY DEFINITE INTEGRALS. 109. We have seen that whenever the indefinite integration l^>(#)cfe can be performed, the value of the definite integral (x, c)dx. a Then u + Su = f 0(o3, & which, by Taylor's theorem, And if z, say, be the greatest value of which be capable, and vanishes in the limit when Sc is indefinitely diminished. Thus in the limit = ^ ' 'dx. a MISCELLANEOUS METHODS AND EXAMPLES. 129 111. The case in which the limits a and b also contain c is somewhat beyond the scope of the present volume. 112. This proposition may be used to deduce many new integrations when one has been performed. Thus since f -- L= =dx = -* tan- 1 \l2=2 (c + a> 0), J (x+cyJx-a Vc + a * c + a we have, by differentiating n times with regard to c, Also, differentiating n times with regard to a, we obtain /- ^ (^ + c)(^ a) 2 Similarly, differentiating this latter p times with regard to c, we obtain r o?^? I- 2^+1 J (^+c)^ +1 (^-a) 2 EXAMPLES. 1. Obtain the following integrals : (i.) f(i+*)-V*fo. (v.) f J J (ii.) An-^-xi + ar)-*^ (vi.) I J J (iii.) r#- 1 (2-3a?+ *?)"*<&. -(vii. E. T. C. 130 INTEGRA L CA LCUL US. 2. Integrate (i.) (a 2 + 6 2 - ^ 2 )v/(a 2 - ^ 2 )(^ 2 - 6 2 )' [ST. JOHN'S, 1888.] 1 (x 2 + a 2 )^^+^ I~ ST - J HN ' S > 1889.] |UL * sin 6>Vacos 2 ^ + 67iii 2 <9+V [TRINITY, 1888.] 3. Find the values of /- ^ f sin x dx J (cos x + cos a)V(cos x 4- cos fi)(cos x + cos y ) /" \ I cLx J cotfx + a\Jcos(a;+B)coa(a: + 'v} C% 1890-] a)\/cos(^ + ^)cos(^ + y } 4. Prove that, with certain limitations on the values of the constants involved, d,L \^ Olll . (x-p)(ax? + Zbx + cy* (-ap 2 -2bp-cft (x - p)(b 2 - acf [TRINITY, 188G ] 5. Prove that \(cQ$x} n dx may be expressed by the series j\r _L pf p ^.v 3 - -r ... etu, n- ND N 2J Nft . . . being the coefficients of the expansion (1 + a) 2 , and n having any real value positive or negative. [SMITH'S PRIZE, 1876.] 6. Evaluate the following definite integrals : (i \ f l W J l /a ^2 /y.2 (a 2 + ^7 2 ) 2 ^' [ST. JOHN'S, 1888.] o / \ f ** x dx ^ UL ' J o (l+tfX 2 +^)(3 + #) [OXFORD, 1888.] 7. Prove that f - 8. Show that [0x^,1888.] MISCELLANEOUS METHODS AND EXAMPLES. 131 9. Evaluate (i. (ii.) | (iii.) 2 + cos x * dx , 4 + 5 sin x [I. C. S., 1839.] o [I. 0. S., 1888.] 10. Prove that / CQ& n xdx is equal to zero o according as n is odd or even. If S denote the sum of the infinite series P rOV6that ^"T-f- [OXFOBD, 1890.] 11. Prove that if c be < 1, (i.) f 8in-^ (ii.) 12. Prove that f^f. . J Vs 13. Find a reduction formula for / e~ T sin [ST. JOHN'S, 1888.] ft 14. Evaluate (i.) / sin x log sin x dx. J [8, 188,5. J o HF (ii.) / tan a? log sin a? o?^?. , ^ ooon k y J [ST. JOHN'S, 1882.] r f (iii.) J s 132 INTEGRAL CALCULUS. 15. Evaluate (i.) f o dx [I. C. S., 1887.] [I. C. S., 1891.] 16. Prove (i.) f^an^^ J sec 37+ cos # 4 [POISSON.] J a 2 cos 2 .2 a being supposed greater than unity. [OXFORD, 1890.] 17. Prove (i.) f 1 -2Sfc ? = - g o 18. Prove that = a a 3 + " & 5 - - ctf + ... z 2 ) 3 3.5 3.5.7 [OXFORD, 1889.] 19. Prove that IT 2r~% /-7A 1 2 1 2 Q2 1 2 o2 PL2 / <^" _T . 1 o 1*3-4 i 1 ^> & x,6 i being supposed < 1. 20. Prove that [MATH. TRIPOS, 1878.] 21. Prove that 1.1 1 ''* [A 1888.] 22. If 23. Prove that -,), \* a $(x)dx= - F $(x)dx. [TRIN. HALL, etc., 1886.] b *"" 6 ^ f^ C ~^W provided c-a?) J ^-6) remains finite when x vanishes. [ST. JOHN'S, 1883.] MISCELLANEOUS METHODS AND EXAMPLES. 133 24. Prove that r a $(x)dx= /""{<##) + <(2a-a?)}cfci7, and illus- * trate the theorem geometrically. 25. If f(x)=f(a+x\ show that and illustrate geometrically. 26. Show that q-pj \ q-p q-p 27. Determine by integration the limiting value of the sums of the following series when n is indefinitely great : ' n + I n + 2 n + 3 n + ri [a, 1884.] / x n n n n (iiL)_J_ + *_+ * .-+ + -J- \/2?i-l 2 \/4?i-2 2 \/6rc-3 2 */2ri*-4 o 4 = ' [circ. of circle], as far as terms involving e 4 . 64 EXAMPLES. 1. Find by integration the length of the arc of the circle = a 2 , intercepted between the points where xa cos a and 2. Show that in the catenary y = c cosh - the length of arc from the vertex (where #=0) to any point is given by . i x s = c smh -. c v 3. In the evolute of a parabola, viz., 4(# - 2a) 3 = 27a# 2 , show that the length of the curve from its cusp (# = 2a) to the point where it meets the parabola is 2a(3v3 1). 4. Show that the length of the arc of the cycloid, .r=a(0 + sin 0) ^ y = a(l-cos<9)J between the points for which 0=0 and 0=2^, is s = 5. Show that in the epicycloid for which y=(a + 6)sin - b sin = a 26 5 beiijg measured from the point at which 0=7rb/a. n 222 When &=--, show that 4r+y*a*j and that if 5 be measured from a cusp which lies on the y-axis, s 3 oc x*. 142 INTEGRAL CALCULUS. 6. Show that in the ellipse # = a cos $, j/ = 6sin^, the peri- meter may be expressed as 7. Find the length of any arc of the curves (i.) r = acos0. (iii.) r = a6. (ii.) r = ae m0 . (iv.) r = asin 2 -. 2t 8. Apply the formula s=_ + \pdty to rectify the cardioide whose equation is r=a(l -f cos 0). [TRINITY, 1888.} 9. Two radii vectores OP, OQ of the curve are drawn equally inclined to the initial line ; prove that the length of the intercepted arc is act, where a is the circular measure of the angle POQ. [ASPARAGUS, Educ. Times.'] 10. Show that the length of an arc of the curve y n =x m+n can be found in finite terms in the cases when or + - is an integer. * m * m * 11. Find the length of the arc between two consecutive cusps of the curve (c 2 a 2 )p 2 = c 2 (r 2 a 2 ). 12. Find the whole length of the loop of the curve 3ay 2 =x(x-a) 2 i . [OXFORD, 1889.] 13. Show that the length of the arc of the hyperbola xy = a? between the limits x=b and x=c is equal to the arc of the curve > 2 (a 4 +r 4 ) = aV 2 between the limits r=b, r=c. [OXFORD, 1888.] 14. Show that in the parabola =1+0080,-^ =:-__^_ and T ' d>Y sin^w* hence show that the arc intercepted between the_yertex and the extremity of the latus rectum is a{\/2-flog(l +\/2)}. [I. C. S., 1882.] RECTIFICATION, ETC. 143 121. Length of the Arc of an E volute. It has been shown (Diff. Gale, for Beg., Art. 157) that the difference between the radii of curvature at Fig. 12. two points of a curve is equal to the length of the corresponding arc of the evolute ; i.e., if ah be the arc of the evolute of the portion AH of the original curve, then (Fig. 12) (at A) /> (at H), .e. 144 INTEGRAL CALCULUS. and if the e volute be regarded as a rigid curve, and a string be unwound from it, being kept tight, then the points of the unwinding string describe a system of parallel curves one of which is the original curve AH. Ex. Find the length of the evolute of the ellipse. Let a, a', /3, /3' be the centres of curvature corresponding to the extremities of the axes, viz., A, A', B, B' respectively. The arc a/3 of the evolute corresponds to the arc AB of the curve, and we have (Fig. 13) arc a/2 = /o(at 5)-p(at A) = ~- [for rad. of curv. of ellipse = ^. Ex. 3, p. 153, Diff. Calc.for Beg."]. Thus the length of the entire perimeter of the evolute EXAMPLE. Show in the above manner for the parabola y 2 = kax that the length of the part of the evolute intercepted within the parabola is4a(3\/3-l). 122. Intrinsic Equation. The relation between s, the length of the arc of a given curve, measured from a given fixed point on Fig. 14. the curve, and the angle between the tangents at the extremities of the arc is called the Intrinsic Equation of the curve. RECTIFICATION, ETC. 145 123. To obtain the Intrinsic Equation from the Cartesian. Let the equation of the curve be given as y=f(x). Suppose the #-axis to be a tangent at the origin, and the length of the arc to be measured from the origin. Then tan -^ =/(), (1) also s=\ *Jl + [f'(x)~] 2 dx (2) If s be determined by integration from (2), and x eliminated between this result and equation (1), the required relation between s and ^ will be obtained. . Ex. 1. Intrinsic equation of a circle. If i/r be the angle between the initial tangent at A and the tangent at the point P, and a the radius of the circle, we have and therefore sa^r. Ex. 2. In the case of the catenary y + c = ccosh-, the in- trinsic equation is s = c tan ^. c For tan^ = ^ dx T as /-, i <>x and = \/ 1 -r smh 2 - = dx > c P. I. c, K 146 INTEGRAL CALCUL US. and therefore s = c sinh -, c the constant of integration being chosen so that x and s vanish together, whence s = c tan \js, 124. To obtain the Intrinsic Equation from the Polar. Fig. 16. / >, Take the initial line parallel to the tangent at the point from which the arc is measured. Then with the usual notation we have T =/(0), the equation to the curve, ....... (1) ^ = 0+0, ....................................... (2) If s be found by integration from (4), and 0, eliminated by means of equations (2) and (3), the required relation between s and \fs will be found. Ex. Find the intrinsic equation of the cardioide r=a(l -cos 0). Here i/r and a sin 2 RECTIFICATION, ETC. 147 Hence and Also and Fig. 17. a sin -, r\ 4a cos - + C. 2 If we determine C so that s = when 9 = 0, we have or 5 = 4a( 1 cos^ J, the intrinsic equation sought. We may notice that if A be the vertex, the arc AP is 4a cos . 148 INTEGRAL CALCULUS, 125. When the Equation of the Curve is given as dy d)'(t) ,- N we have tan ^ = -^ = ^_-^ ....... ..... ........ (1) ax j (t) By means of equation (2) s may be found by in- tegration in terms of t. If then, between the result and equation (1) t be eliminated, we shall obtain the required relation between s and ^. Ex. In the cycloid ya(\ -cos t\ we have tan ^ = , * mt = tan * 1+costf 2 Also ^ = ax/(l + cos0 2 + sin 2 * = 2a cos -, dt 2 whence 5=4a sin - if s he measured from the origin where Z=0. Hence 5 = 4ot sin T/T is the equation required. 126. Intrinsic Equation of the Evolute. Let s=f(\f/) be the equation of the given curve. Let s' be the length of the arc of the evolute measured from some fixed point A to any other point Q. Let and P be the points on the original curve corre- sponding to the points A, Q on the evolute; p , p the radii of curvature at and P: \j/ the angle the tangent QP makes with OA produced, and ^ the angle the tangent PT makes with the tangent at 0. RECTIFICATION, ETC. Then *// = -^r, and ds 149 or O T * Fig. 18. 127. Intrinsic Equation of an Involute. With the same figure, if the curve AQ be given by the equation s'=f(\}/), we have and \Is \//, whence = \ Ex. The intrinsic equation of the catenary is s=ctsm\Ir (Art. 123). Hence the intrinsic equation of its evolute is and p = radius of curvature at the vertex = c p = y = c sec 2 i/r and T/r=0 , . ' . the evolute is s = c(sec 2- v//- - 1 ), or s = c tan 2 ^. The intrinsic equation of an involute is s = I (c tan "fy + A)d^r = c log sec T/T + A^r + constant ; and if s be so measured that 5=0 when ^=0, we have s = c log(sec T 150 INTEGRAL CALCULUS. 128. Length of Arc of Pedal Curve. If p be the perpendicular from the origin upon the tangent to any curve, and ^ the angle it makes with the initial line, we may regard p, % as the current polar coordinates of a point on the pedal curve. Hence the length of the pedal curve may be cal- culated by the formula Ex. Apply the above method to find the length of any arc of the pedal of a circle with regard to a point on the circum- ference (i.e. a cardioide). Fig. 19. Here, if 2a be the diameter, we have from the figure p = OP cos * = 2ctcos 2 *. Hence arc of pedal = / 2 A/a 2 cos 4 - + a 2 sin 2 - J * 2 2 = / j - cos -a 2 2a cos *dx = 4a sin + C. The limits for the upper half of the curve are x = and X = TT. Hence the whole perimeter of the pedal 2[4asin- 1 =8a. L- 2 Jo RECTIFICATION, ETC. 151 EXAMPLES. 1-. Find the length of any arc of the curve f u\a x)=aP. [a, 1888.] 2. Find the length of the complete cycloid given by y = a a cos 0. 1 3. Find the curve for which the length of the arc measured from the origin varies as the square root of the ordinate. 4. Show that the intrinsic equation of the parabola is s = a tan ^ sec ^r + a log(tan i/r + sec ^). 5. Interpret the expressions wherein the line integrals are taken rou id the perimeter of a given closed curve. [ST. JOHN'S, 1890.] Jw 6. The major axis of an ellipse is 1 foot in length, and its ^/^eccentricity is 1/10. Prove its circumference to be 3*1337 feet nearly. [TRINITY, 1883.] 7. Show that the length of the arc of that part of the cardioide r = a(l + cos 0), which lies on the side of the line 4r=3asec remote from the pole, is equal to 4a. [OXFORD, 1888.] 8. Find the length of an arc of the cissoid r _ a sin 2 6> cos ff 9. Find the length of any arc of the curve 10. Show that the intrinsic equation of the semicubical para- bola 3a3/ 2 =2^ is 9s = 4a(sec 3 Vr - 1). 11. In a certain curve show that 5=e e \/2 + a 152 INTEGRAL CALCULUS. 12. Show that the length of an arc of the curve is given by s =/(<9) +/"(#) + C. 13. Show that in the curve y = alogsec- the intrinsic equa- tion is s = a gd~ l \ff. a 14. Show that the length of the arc of the curve y=logcoth- between the points (x l9 yj), (# 2 , 3/2) is log s ! n x ^. sinn X-^ 15. Trace the curve y 2 = g (a #) 2 , and find the length of that od/ part of the evolute which corresponds to the loop. [ST. JOHN'S, 1881 and 1891.] 16. Find the length of an arc of an equiangular spiral (p = rsma) measured from the pole. Show that the arcs of an equiangular spiral measured from the pole to the different points of its intersection with another equiangular spiral having the same pole but a different angle will form a series in geometrical progression. [TRINITY, 1884.] 17. Show that the curve whose pedal equation is p 2 =r 2 a? has for its intrinsic equation s = a-. Zi 18. Show that the whole length of the limagon r=acos is equal to that of an ellipse whose semi-axes are equal in length to the maximum and minimum radii vectores of the limacon. 19. Prove that the length of the nth pedal of a loop of the curve r m =a m sinmO is ,-m mn-m+1 a(mn+I) (smmO) m dO. ^ 1883 j o 20. Show that the length of a loop of the curve [ST. JOHN'S, 1881.] CHAPTER X. QUADRATURE, ETC. 129. Areas. Cartesians. The process of finding the area bounded by any portion of a curve is termed quadrature. It has been already shown in Art. 2 that the area bounded by any curved line [y = (x)], any pair of ordinates [x = a and x = b] and the axis of x 9 may be considered as the limit of the sum of an infinite num- ber of inscribed rectangles; and that the expression for the area is 1 ydx or (x)dx. In the same way the area bounded by any curve, two given abscissae [y = c, y = d] and the y-axis is f xdy. 130. Again, if the area desired be bounded by two given curves [y = (0) - \fs(x)]dx. x=a J Fig. 20. Ex. 1. Find the area bounded by the ellipse - + ^- = 1, the 2 2 ordinates x=c, xd and the Here area= f ^Sr J a - b 2 2a For a quadrant of the ellipse we must put d =a and c=0 and the above expression becomes . a 2 . ? or ^ . 2a 2 4 giving irab for the area of the whole ellipse. Ex. 2. Find the area above the #-axis included between the curves y 2 = %ax x 2 and y 2 = ax. The circle and the parabola touch at the origin and cut again at (a, a). So the limits of integration are from #=0 to xa. The area sought is therefore f a ? - x 2 ~ QUADRATURE, ETC. Now, putting x=a(\ cos 0\ 155 2 1 TT _ira 6 2 2~~4~' and / J~axdx = *Ja\ = fa 2 . J Lf Jo Thus the area required is a?( j J. Fig. 21. Ex. 3. Find the area (1) of the loop of the curve (2) of the portion bounded by the curve and its asymptote. Here To trace this curve we observe : (1) It is symmetrical about the ^7-axis. (2) No real part exists for points at which x is > a or <-a. (3) It has an asymptote #+a=0. (4) It goes through the origin, and the tangents there are y x " ,7 (5) It crosses the #-axis where x a^ and at this point -f- is infinite. dx (6) The shape of the curve is therefore that shown in the figure (Fig. 22). Hence for the loop the limits of integration are to a, and then double the result so as to include the portion below the 156 INTEGRAL CALCULUS. For the portion between the curve and the asymptote the limits are a to 0, and double as before. For the loop we therefore have a+x for the portion between the curve and the asymptote, /O In _ /v. x\l dx. > a+x Fig. 22. To integrate I xJ a ~ x dx, put J * a ~p x x=a cos and dx a sin c?$. ftfJEfcfc- - l\ J Va+x J Then 1-cos 2 ^ r = a 2 / and area of --} I) QUADRATURE, ETC. 157 Again, r x J?E*dx= - a cos fl^ 1 "^gain Bd9 J_ >a+# J > l-cos 2 # [The meaning of the negative sign is this : In choosing the + sign before the radical in y=#/v/^ _ we are tracing the * a+x portion of the curve below the #-axis on the left of the origin and above the axis on the right of the origin. Hence y being negative between the limits referred to, it is to be expected that we should obtain a negative value for the expression Thus the whole area required is [It must also be observed in this example that the greatest ordinate is an infinite one. In Art. 2 it was assumed that every ordinate was finite. Is then the result for the area bounded by the curve and the asymptote rigorously true ? To examine this more closely let us integrate between limits a + e and 0, where e is some small positive quantity, so as to exclude the infinite ordinate at the point x a, we have as before [ is the angle between the tangent and the radius vector) 137. Tangential-Polar Form. ds . = 5 ^ = P + = \ \p ds = J ds . d*p Again, since P = = P + we have area QUADRATURE, ETC. 165 a formula suitable for use when the Tangential-Polar equation is given. 138. Closed Curve. When the curve is closed this expression admits of some simplification. For and in integrating round the whole perimeter the first term disappears. Hence when the curve is dosed we have area = ^ Ex. By Ex. 23, p. 113, Diff. C ale. for Beginners, the equation of the one-cusped epicycloid (i.e., the cardioide) may be expressed as p = Fig. 29. Hence its whole area=-^ / f 9a 2 sin 2 ;' a 2 cos 2 ^ \d^ taken be- tween limits i/r = and ^= and doubled. Putting -^ = 3$, this becomes IT = 3a 2 f (9 sin 2 ^ - co**0)dO = 67ra 2 . ^o 166 INTEGRAL CALCULUS. 139. Pedal Equation. Again, for curves given by their pedal equations, we have A = ip ds = i p dr = }p sec dr = i - Ex. In the equiangular spiral p=r sin a, Hence any sectorial area /> 2 y2 s i = f / J r rcosa 140. Area included between a curve, two radii of curvature and the evolute. In this case we take as our element of area the elementary triangle contained by two contiguous radii of curvature and the infinitesimal arc ds of the curve. To first order infinitesimals this is |/o 2 F 1 (=^ 1 ) the per- pendicular from P upon the tangent. Then the areas 170 INTEGRAL CALCULUS. of the pedals with and P respectively as origins are ijV 2 cfyr and j[ft L 2 <% taken between certain definite limits. Call these areas A and A l respectively Let r, 6 be the polar coordinates of P with regard to 0, and x 9 y their Cartesian equivalents. Then Pi P ~~ T cos($ -t/r) =p x cos ifs y sin i/r, and p is a known function of \fs Hence 2 A l = \p^d\fr = \(p x cos i/r y sin \l/fd\^ = Vp^d"^ 2x \p cos ^ d\/r 2^/ |p sin \fs d\fs + x* I cos 2 i/r c?i/r + 2a32/ 1 cos \[s si + 2/- 2 1 si 2 Jp cos \/r d\fs, 2 Ip sin \/r d\[s, I Now between such limits that the whole pedal is described will be definite constants. Call them -20, -2/, a, 2A, 6, and we thus obtain 2A l = < 2A + 2gx + 2fy + ax 2 + 2hxy + by 2 . If then P move in such a manner that A l is constant, its locus must be a conic section. 143. Character of Conic, It is a known result in inequalities that Hence it will be obvious that if p, q, r, ... , stand for QUADRATURE, ETC. 171 cos/^, eos2h, cos3/i, ... , cosnh, and p v q v r v ... , for sin A, sin 2/z,, etc., we shall have in the limit when h is made indefinitely small and nh finite = ^, say, I cos 2 \fsd\[sX I sin 2 !//- c?^ > / 1 sin\/r cosi/r cZx/^J , oo o i.e. ab>h 2 . Hence our conic section is in general an ellipse. Moreover the position of its centre is given by ax+hy+g = 0*\ hx+by+f=0j and is independent of the magnitude of A v Hence for different values of A l these several conic-loci will all be concentric, We shall call this centre 1 144. Closed Oval. Next suppose that our original curve is a closed oval curve, and that the point P is within it. Then the limits of integration are and 2?r. p2* p2ir Thus a = cos 2 ^ d\js = TT = I sin 2 \/r d\{t = b o o and h = cos \/r sin i/r d^/r = 0. o Hence the conic becomes i.e. a circle whose centre is at the point ^ p27T ^ /2 I pcosi/rcZ^, - I p sin \fsd\fs. 7TJ 7TJ 145. Connexion of Areas. The point 2 having been found, let us transfer our origin from to 2. The linear terms of the conic 172 INTEGRAL CALCULUS, will thereby be removed. Thus 2 is a point such that the integrals \pcos\fsd\fs and \psiu\^d\[^ both vanish, and if II be the area of the pedal whose pole is 2 we have for any other 2A 1 = 211 + ax 2 + 2hxy + by* in the general case. The area of this conic is * (Smith's Conic Sections, Art. 171). Thus A TT , */ab h 2 , I? \ -d. 1 = IH -- s "" (area or conic). ^7T For the particular case of any closed oval the equa- tion of the conic becomes whence J. 1 where r is the radius of the circle on which P lies f o r constant values of A v i.e. the distance of P from 2. 146. Position of the Point 2 for Centric Oval. In any oval which has a centre the point 2 is plainly at that centre, for when the centre is taken as origin, the integrals \pcos\fsd\fs and \psui\fsd\fs both vanish when the integration is performed for the complete oval (opposite elements of the integration cancelling), 147- Ex. 1. Find the area of the pedal of a circle with regard to any point within the circle at a distance c from the centre (a limagon). Here A = n+^, and n = 7ra 2 . Hence Ai=ica*+ . QUADRATURE, ETC. 173 Ex. 2. Find the area of the pedal of an ellipse with regard to any point at distance c from the centre. In this case II is the area of the pedal with regard to the centre - 2 /* Vcos 2 <9 + b%m*0)dO = (a 2 + 2 )|. Hence ^ 1 =|( Ex. 3. The area of the pedal of the cardioide r=a(l cos 0) taken with respect to an internal point on the axis at a distance c from the pole is |(5s- 2 c + 2c'). [MATH . TBIpos>187a] Let be the pole, P the given internal point ; p and p l the two perpendiculars OF 2 and PT l on any tangent from and P respectively ; < the angle Y$P and OPc ; then p l p c cos <, and ^A l =2A 2clp cos < cfc + / Fig. 36. Now in order that p may sweep out the whole pedal we must integrate between limits < = and < = -^ and double. Now in the cardioide (Fig. 36) p= OQ sin Y 2 QO = OQ sin^xOQ. [Dif. Calc., p. 190.] 174 INTEGRAL CALCULUS. For r 2 = itf0 = | Hence |-{*-(-W-| or |-*=f, - J-J-J, /3 so - -. 23 , A*- , Hence / p cos d = 2 / 2a cos 3 2 cos d(f> ' fl = 4a x 3 / cos% cos 3,s o?2 o r f = 12# / [4 cos% - 3 cos)cta=3.2c 2 i ^^^ J 222 Sir Finally 24 = 2 J 4a**^*J> = 24a 2 Tcos^ TcoW [I. C. S., 1881] In what ratio does the line x+y = Za divide the area of the loop? [OXFORD, 1889.] 22. Find the area of the curve r=aOe be enclosed between two given radii vectores and two successive branches of the curve. [TRINITY, 1881.] 23. Find the area of the loop of the curve r = a0cos between 24. Show that the area of a loop of the curve r = acosn0 is ^ -, and state the total area in the cases n odd, n even. 4?i 25. Find the area of a loop of the curve r = a cos 3$ + b sin 3$. [I. C. S., 1890.] 26. Show that the area contained between the circle r=a and the curve r=acos5$ is equal to three-fourths of the area of the circle. [OXFORD, 1888.] 27. Prove that the area of the curve r 2 (2c 2 cos 2 <9 - 2ac sin cos 9 + a 2 sin 2 #) = aV is equal to irac. [I. C. S., 1879.] 28. Find the whole area of the curve represented by the equation r = acos + b, assuming b > a. 29. Find the area included between the two loops of the curve r = a(2 cos + ^3). [OXFORD, 1889. ] 30. Find the area between the curve r=a(sec $+cos 0) and its asymptote. 31. Prove that the area of one loop of the pedal of the lemniscate r 2 = a 2 cos2$ with respect to the pole is a 2 . [OXFORD, 1885.] 32. Find the area of the loop of the curve (x'\-y)(x^+y 2 ) = ^axy. [OXFORD, 1890.] 33. Prove that the area of the loop of the curve QUADRATURE, ETC. 181 34. Find the whole area contained between the curve and its asymptotes. [OXFORD, 1888.] 35. Show that the area of the ellipse ( *L^ = a 2 +b 2 -r 2 in- p 2 eluded between the curve, the semi-major axis, and a radius vector r from the centre, is tan" 1 ^/^^-, a, b being the semi-axes of the ellipse. [CLARE, etc., 1882.] 36. Show that the area in eluded between the curve 5 = atan^, its tangent at ^ = and its tangent at V*" <> i g -a 2 tan + a 2 tan ^ - a 2 log(sec < + tan c). [TRINITY, 1892.] 37. Show that the area of the space between the epicycloid p =^isin Sty and its pedal curve taken from cusp to cusp is ^irA 2 B. 38. Show that the curve r = a(^\/3 + cos^#) has three loops whose areas are a 2 (f TT + 2\/3), a 2 (f TT - f\/3), a\ -far - f V) re- spectively. [COLLEGES, 1892.] 39. Find the area of a loop of the curve x*+y* = Za 2 xy. [OXFORD, 1888.] 40. Find the area of the pedal of the curve =*(<**- d*)l, the origin being taken at x = *Ja 2 6 2 , y = 0. [OXFORD, 1888.] 41. Find the area included between one of the branches of the curve 3% 2 = a 2 (# 2 +;?/ 2 ) and its asymptotes. [a, 1887.] 42. Find the whole area of the curve tf+yi = a\x*+y*). [a, 1887.] 43. Find the area of a loop of the curve (mV + n,y)* = aV - b 2 y 2 - [ST. JOHN'S, 1887. ] 44. Trace the shape of the following curves, and find their areas : (i.) (^+^2)3 =aa y*. (ii.) (^ 2 + 2/) 3 = a^?/ 4 . [BELL, etc., SCHOLARSHIPS, 1887.] 45. Prove that the area of 3? V 2 1 / ' X 2 V 2 \ 2 7TC 2 / ' 182 INTEGRAL CALCULUS. 46. Prove that the area in the positive quadrant of the curve (av+w^w is ^(5+5). [a;18900 47. Prove that the area of the curve f") is - 3 & + (V 2 - a 2 ) tan- 1 - . [ST. JOHN'S, 1883.] 48. Prove that the area of the curve 9 ,aV h 6 2 where c is less than both a and 5, is 7r(ab c 2 ). [OXFORD, 1890.] 49. Prove that the area of the curve ^ 4 -3o^ 3 + a 2 (2^ 2 +y 2 )=0 is fTrtt 2 . [MATH. TRIPOS, 1893.] 50. Prove that the areas of the two loops of the curve [MATH. TRIPOS, 1875.] are (32^ + 24^3) a 2 , and (167r-24\/3)a 2 , CHAPTER XI. SUKFACES AND VOLUMES OF SOLIDS OF KEVOLUTION. 153. Volumes of Revolution about the a>axis. It was shown in Art. 5 that if the curve y=f(x) revolve about the axis of x the portion between the ordinates x = x^ and x = x 2 is to be obtained by the formula *2 Tr 2 . dx. 154. About any axis. More generally, if the revolution be about any line AB, and if PN be any perpendicular drawn from a 184 INTEGRAL CALCULUS. point P on the curve upon the line AB and P'N' a contiguous perpendicular, the volume is expressed as or if be a given point on the line AB 155. Ex. 1. Find the volume formed by the revolution of the loop of the curve f=x 2 ?^ (Art. 130, Ex. 3) about the tf-axis. Here volume =/ 7ry 2 dx=7r I x* a ~ x dx. J J J a + x o o Putting a +x=z, this becomes rf2a 3 log z - 5a 2 z + 2az 2 - ~ 3 _J Ex. 2. Find the volume of the spindle formed by the revolu- tion of a parabolic arc about the line joining the vertex to one extremity of the latus rectum. Fig. 40. Let the parabola be y 2 = 4o Then the axis of revolution is y = 2^7, and P fi VOLUME OF REVOLUTION. 185 Also and volume = . dAN o 4?r out 75 156. Surfaces of Revolution. Aain, if S be the curved surface of the solid traced y the revolution of any arc AB about the ^c-axis, Fig. 41. suppose PN, QM two adjacent ordinates, PN being the smaller, 3s the elementary arc PQ, SS the area of the elementary zone traced out by the revolution of PQ 186 INTEGRAL CALCULUS. about the #-axis, y and y + Sy the lengths of the ordinates of P and Q. Now we may take it as axiomatic that the area traced out by PQ in its revolution is greater than it would be if each point of it were at the distance PN from the axis, and less than if each point were at a distance QM from the axis. Then SS lies between ^y 8s and 2w(y + Sy)Ss, and therefore in the limit we have r/^ f -j- = 2-7T2/ or 8 \ This may be written as as may happen to be convenient in any particular example, the values of -r-> -j-> -^, etc., being obtained from the differential calculus. 157. Ex. 1. Find the surface of a belt of the paraboloid formed by the revolution of the curve y 2 = &ax about the #-axis. Here = dx V x dx /X<1 / ydx dx Ex. 2. The curve r = a(l + cos 6} revolves about the initial line. Find the volume and surface of the figure formed. Here volume = / try^dx = TT / ?' 2 sin 2 d(r cos 0) = TT / a' 2 (l -{- cos #) 2 sin 2 #a c/(cos -f cos 2 $), S URFA CE OF RE VOL UTION. 187 the limits being such that the radius sweeps over the upper half of the curve. Hence volume - - 7ra 3 T(l + cos 0) 2 (1 + 2 cos 0)sin :j dO *o 3 T(l + 4 cos + 5 cos 2 + 2 cos 3 0)sin W0 (l 4-5 cos 2 0)sin 3 Fig. 42. The surface = 2ir \yds=2ir T r sin 0~c?0 J J dO = 2?r / ^ a(l + cos (9)sin <9\/a 2 (l + cos 0) 2 27ra 2 (l + cos 0)sin (9 . 2 cos o 4 16?ra 2 / ^ cos 4 o 2 2 e EXAMPLES. 1. Obtain the surface of a sphere of radius a (i.) by Cartesians, (ii.) by polars, taking the origin on the circumference. 188 INTEGRAL CALCULUS. 2. A quadrant of a circle, of radius a, revolves round its chord. Show that the surface of the spindle generated and that its volume = -^-(10 - 3?r). 3. The part of the parabola y L = ax cut off by the latus rectum revolves about the tangent at the vertex. Find the curved surface and the volume of the reel thus generated. THEOREMS OF PAPPUS OR GULDIN. 158. I. When any closed curve revolves about a line in its own plane, which does not cut the curve, the volume of the ring formed is equal to that of a cylinder whose base is the curve and whose height is the length of the path of the centroid of the area of the curve. Let the #-axis be the axis of rotation. Divide the area (A) up into infinitesimal rectangular elements with sides parallel to the coordinate axes, such as Fig. 43. PjPgPgP^, each of area SA. Let the ordinate P l N l = y. Let rotation take place through an infinitesimal angle 89. Then the elementary solid formed is on base SA and its height to first order infinitesimals is ySO, and therefore to infinitesimals of the third order its volume is SA . THEOREMS OF PAPPUS. 189 If the rotation be through any finite angle a we obtain by summation SA . y . a. If this be integrated over the whole area of the curve we have for the volume of the solid formed a!i/cL4. Now the formula for the ordinate of the centroid of a number of masses m v m 2 , ..., with ordinates X?7? II 2/i> 2/2' > i s y = y -^ then we seek the value of the ordinate of centroid of the area of the curve, each element 8A is to be multiplied by its ordinate and the sum of all such products formed, and divided by the sum of the elements, and we have or in the language of the Integral Calculus (yd A (yd A y = J _ = i - . \dA A Thus Therefore volume formed = A(ciy). But A is the area of the revolving figure and ay is the length of the path of its centroid. This establishes the theorem. COR. If the curve perform a complete revolution, and form a solid ring, we have a = 2-7T and volume = A(2jry). 159. II. When any closed curve revolves about a line in its own plane which does not cut the curve, the curved surface of the ring formed is equal to that 190 INTEGRAL CALCULUS. of the cylinder whose base is the curve and whose height is the length of the path of the centroid of the perimeter of the curve. Let the #-axis be the axis of rotation. Divide the perimeter s up into infinitesimal elements such as P X P 2 each of length Ss. Let the ordinate P l N l be called y. Let rotation take place through an infinitesimal angle S9. Then the elementary area formed is ultimately a rectangle with sides Ss and ySO, and to infinitesimals of the second order its area is Ss . y9. Fig. 44. If the rotation be through any finite angle a we obtain by summation Ss . ya. If this be integrated over the whole perimeter of the curve we have for the curved surface of the solid formed an/cfe. If we seek the value of the ordinate (rj) of the centroid of the perimeter of the curve, each element Ss is to be multiplied by its ordinate, and the sum of THEOREMS OF PAPP US. 191 all such products formed, and divided by the sum of the elements, and we have Lt or in the language of the Integral Calculus ^yds \yds n \ds s Thus \yd8=8tj, and the surface formed = s(afj). But s is the perimeter of the revolving figure, and arj is the length of the path of the centroid of the perimeter. This establishes the theorem. Con. If the curve perform a complete revolution and form a solid ring, we have a = 2?r and surface = s( 2 -73-77). Ex. The volume and surface of an anchor-ring formed by the revolution of a circle of radius a about a line in the plane of the circle at distance d from the centre are respectively volume = Tra 2 X 2?rc? = 27T 2 a 2 o?, surface = 2:ra x Zird = 4ir 2 ad. EXAMPLES. 1. An ellipse revolves about the tangent at the end of the major axis. Find the volume of the surface formed. 2. A square revolves about a parallel to a diagonal through an extremity of the other diagonal. Find the surface and volume formed. 3. A scalene triangle revolves about any line in its plane which does not cut the triangle. Find expressions for the surface and volume of the solid thus formed. 192 INTEGRAL CALCULUS. 160. Revolution of a Sectorial Area. When any sectorial area OAB revolves about the initial line we may divide the revolving area up into infinitesimal sectorial elements such as OPQ, whose area may be denoted to first order infinitesimals by |r <2 o0. Being ultimately a triangular element, its centroid is f of the way from along its median, and in a complete revolution the centroid travels a distance 27r(f r sin 6) or f irr sin 9. Fig. 45. Thus by Guldin's first theorem the volume traced by the revolution of this element is to first order infinitesimals, and therefore the volume traced by the revolution of the whole area OAB is f 7r[r 3 si sin 9 d9. 161. If we put x = rcos9, y = rsin9, and = we have r 3 sin 9 S9 = r 3 sin (9$(tan ~ l f) = r 3 sin 9 . 7 = r*cos*9t St = xH St, EXAMPLES. 193 and the volume may therefore be expressed as (xHdt. EXAMPLES. 1. Find by integration the volume and surface of the right circular cone formed by the revolution of a right-angled triangle about a side which contains the right angle. 2. Determine the entire volume of the ellipsoid which is generated by the revolution of an ellipse around its axis major. [I. C. S., 1887.] 3. Prove that the volume of the solid generated by the revolution of an ellipse round its minor axis, is a mean pro- portional between those generated by the revolution of the ellipse and of the auxiliary circle round the major axis. [I. C. S., 1881.] 4. Prove that the surface of the prolate spheroid formed by the revolution of an ellipse of eccentricity e about its major axis is equal to 2 . area of ellipse . Prove also that of all prolate spheroids formed by the revolu- tion of an ellipse of given area, the sphere has the greatest surface. [I. C. S., 1891.] 5. Find the volume of the solid produced by the revolution of the loop of the curve y^x^ about the axis of x. [I. C. S., 1892.] 6. Find the surface and volume of the reel formed by the revolution of the cycloid round a tangent at the vertex 7. Show that the volume of the solid formed by the revolu- tion of the cissoid y 2 (2a ^)=x 3 about its asymptote is equal to 2?r 2 a 3 . [TRINITY, 1886.] 8. Find the volume of the solid formed by the revolution of the curve (a - x)y 2 = a?x about its asymptote. [I. C. S. , 1883. ] 9. If the curve r = a + b cos revolve about the initial line, show that the volume generated is 7ra(a 2 4- b 2 ) provided a be greater than b. [a, 1884.] E. i. c. N 194 INTEGRAL CALCULUS. 10. Find the volume of the solid formed by the revolution about the prime radius of the loop of the curve r^ = between 6 = and 0=|. [O x TO K D , 1890.] 11. Show that if the area lying within the cardioide and without the parabola r(l+cos $) = 2a, revolves about the initial line, the volume generated is 187ra 3 . [TRINITY, 1892.] 12. The loop of the curve Zay 2 =x(x a) 2 revolves about the straight line y=a. Find the volume of the solid generated. [OXFORD, 1890.] 13. Show that the coordinates of the centroid of the sectorial area of r=f(0) bounded by the vectors 0=a, 6 = ft, has for its coordinates f 14. Show that the centroid of the cardioide r = a(l cos $) is on the initial line at a distance - from the origin. 6 15. If the cardioide r = a(l cos #) revolve round the line p=rcos(9 y\ prove that the volume generated is 3^7r% 2 + f 7T 2 3 cos y. [ST. JOHN'S, 1882.] 16. The curve r=a(l -ecos 0\ where e is very small, revolves about a tangent parallel to the initial line. Prove that the volume of the solid thus generated is approximately 27r 2 a 3 (l+e 2 ). [I. C. S., 1892.1 17. The lemniscate r 2 = a 2 cos2# revolves about a tangent at _ the pole. Show that the volume generated is - CHAPTER XII. SURFACE INTEGRALS. SECOND-ORDER ELEMENTS OF AREA. MISCELLANEOUS APPLICATIONS. 162. Use of Second Order Infinitesimals as Ele- ments of Area. For many purposes it is found necessary to use for our elements of area second order infinitesimals. 163. Suppose, for instance, we desire to find the mass of the area bounded by a given curve, the #-axis, and a pair of ordinates, when there is a distribution of surface-density over the area not uniform, but represented at any point by cr = (x,y)dxdy, the elements dx, dy being written in the reverse order. There is no uniformly accepted convention as to the order to be observed, but as the latter appears to be the more frequently used notation, we shall in the present volume adopt it and write 'x, y)dxdy when we are to consider the first integration to be made with regard to y and the second with regard to x, and when the first integration is with regard to x. That is to say, the right-hand element indicates the first integration. Ex. If the surface-density of a circular disc bounded by xP+y 2 = a 2 be given to vary as the square of the distance from the y-axis, find the mass of the disc. &JL^ ^ Here we have [juv 2 for the .mass of the element 8x 6y, and its mass is therefore /*# 2 y, and the whole mass will be / / The limits for y w411 be ;?/ = to y=*Jd l x L for the positive quadrant, and for x from #=0 to x=a. The result must then 198 INTEGRAL CALCULUS. be multiplied by 4, for the distribution being symmetrical in the four quadrants the mass of the whole is four times that of the first quadrant. Fig. 47. Putting x=asm0 and dx=acosOdO, we have 165. Other Uses of Double Integrals. The same theorem may be used for many other purposes, of which we give a few illustrative examples, which may serve to indicate to the student the field of investigation now open to him. But our scope in the present work does not admit exhaustive treatment of the subjects introduced. DO UBLE INTEGRA TION. 199 Ex. Find the statical moment of a quadrant of the ellipse r 2 4,2 _+_ = ! 9 ' 1,9 a 2 6 2 about the y-axis, the surface-density being supposed uniform. Here each element of area 8x8y is to be multiplied by its surface density cr (which is by hypothesis constant in the case supposed) and by its distance x from the y-axis, and the sum of such elementary quantities is to be found over the whole quadrant. The limits of the integration will be from y=0 to 7 y = _Va 2 x* for ?/ ; and from #=0 to x=a for x. Thus we have a moment = / / crxdxd'u=^\ Wa 2 x 2 dx J J a ) 00 _)d<9 Hence x = -?ra 3 / -?ra 2 = -a. Ex. 3. In a circle the surface-density varies as the nfh power of the distance from a point on the circumference. Find the moment of inertia of the area about an axis through perpen- dicular to the plane of the circle. Here, taking for origin and the diameter for initial line, the bounding curve is r=2acos 6, a being the radius. The density =p,r". Hence the mass of the element rSOSr is //,r n+1 S$Sr, and its moment of inertia about the specified axis is //,r n+3 8$ 8r. Hence the moment of inertia of the disc is f ffj where the limits for r are to 2a cos 0, and for 0, to ~ (and double). Thus Mom. Inertia = J?^L(2a)+4 r c os+ 4 (9 dO n + 4 J DOUBLE INTEGRATION. 207 Again, the mass of the disc is r"5" /*2acos0 = 2|J ^o = _?^L(2a) w+2 f r cos w+2 0d0. n + 2 J n Hence Mom. Inertia = 4 EXAMPLES. 1. Find the centroid of the sector of a circle (a) when the surface-density is uniform, (ft) when the surface-density varies as the distance from the centre. 2. Find the centroid of a circle whose surface-density varies as the nth power of the distance from a point on the circum- ference. Also its moments of inertia (1) about the tangent at 0, (2) about the diameter through 0. 3. Show that the moment of inertia of the triangle of uniform surface-density bounded by the ?/-axis and the lines y = m l x+c l ^ y=mtfc + c 2/i> 2/2> a > &) = > ( 3 ) and from these three equations a and b may theoreti- cally be eliminated (if they have not already dis- appeared by the process of differentiation), and there will result a relation connecting x, y, y v y 2 ; say F(x>y, yi>y z ) => the differential equation of the family. 173. Order of an Equation. We define the order of a differential equation to be the order of the highest differential coefficient occurring in it ; and we have seen that if an equation between 214 DIFFERENTIAL EQUATIONS. two unknowns contains one arbitrary constant the result of eliminating that constant is a differential equation of the first order; and if it contain two arbitrary constants the result is a differential equation of the second order. And our argument is general : so that to eliminate n arbitrary constants we shall have to proceed to n differentiations, and the result is a differential equation connecting x, y, y v ...,2/ n > an( l is therefore of the nth order. Ex. 1. Eliminate a and c from the equation x 2 +y 2 =2ax + c. Differentiating, x -f yy^ = a. Differentiating again, l+^+y^^^ and the constants having disappeared we have obtained as their eliminant a differential equation of the second order (?/ 2 being the highest differential coefficient involved), which belongs to all circles whose centres lie on the #-axis. Ex. 2. Form the differential equation of all central conies whose axes coincide with the axes of coordinates. Here the typical equation of a member of this family of and we have and whence x(y? + yy 2 ) -yy l =0 is the differential equation sought. 174. Elimination an irreversible process. Now this process of elimination is not in general a reversible process, and when we wish to discover the typical equation of a member of a family of curves when the differential equation is given, we are com- pelled to fall back, as in the case of integration, upon a set of standard cases, and many equations may arise which are not solvable at all. We may infer, however, that in attempting to solve a differential equation of the nth order we are to search for an algebraical relation between x, y, and n VARIABLES SEPARABLE. 215 arbitrary constants, such that when these constants are eliminated the given differential equation will result. Such a solution is regarded as the most general solution obtainable. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 175. There are five standard forms. CASE I. Variables Separable. All equations in which it is possible to get dx and all the x's to one side, and dy and all the y's to the other, come under this head, and solve immediately by integration. Ex.1, Thus if sec y= sec x-, dx we have cos x dx cos y dy, and integrating, sin x = sin y + A , a relation containing one arbitrary constant A. Ex. 2. If x = xy-^* y+l dx we have ( x + - ) dx = (y 2 + y)dy, \ x J 2 32 and therefore + log x^ +y~- + A, 2 32 containing one arbitrary constant A. EXAMPLES. y Solve the following differential equations : 1. I dy = x*+x+\ dy y*+y+l ' dx *++l' ' dxx*+x+l / 4. Show that every member of the family of curves in Ex. 3 cuts every member of the set in Ex. 2 at right angles. 216 DIFFERENTIAL EQUATIONS. 7. Show that all curves for which the square of the normal is equal to the square of the radius vector are either circles or rectangular hyperbolae. 8. Show that a curve for which the tangent at each point makes a constant angle (a) with the radius vector can belong to no other class than r=Ae^ cot a . 9. Find the equations of the curves for which (1) the Cartesian subtangent is constant, (2) the Cartesian subnormal is constant, (3) the Polar subtangent is constant, (4) the Polar subnormal is constant. 10. Find the Cartesian equation of the curve for which the tangent is of constant length. 176. CASE II. Linear Equations. [DEF. An equation of the form when P, Q, . . . , K, R are functions of x or constants is said to be linear. Its peculiarity lies in the fact that no differential coefficient occurs raised to a power higher than the first.] As we are now discussing equations of the first order, we are limited for the present to the case If this be multiplied throughout by er ' * it will be seen that we may write it d , /Pefccv n fPdx dP e )=<^ Thus ye fPdx =\Qe /Pdx dx+A, a relation between x and y satisfying the given differential equation, and containing an arbitrary constant. It is therefore the solution required. The factor e ' which rendered the left-hand mem- ber of the equation a perfect differential coefficient is called an " integrating factor." LINEAR EQUATIONS. 217 Ex. 1 . Integrate yi+xy = x. fxd~ - Here e-** or e 2 is an integrating factor, and the equation mav be written d - - (ye*)=xe*, ax & *2 or ye*=e*+A, + i.e. y = l+Ae 2 . Ex. 2. Integrate ^l + -y=x 2 . dx x Jldx Here the integrating factor is e x =e logx =x, and the equation may be written *JH-+ x* and xy=--+A, 177. Equations reducible to linear form. Many equations, if not immediately of the linear form _ may be immediately reduced to it by change of the variables. One of the most important cases is that of the equation Or y-n Putting y l ~ n = z, we have y-^dy=^ 218 DIFFERENTIAL EQUATIONS. or which is linear, and its solution is (l-ri)fpdx ,~ \fr\ Q-~ n )f Pdx J ze =(1 ft) \Q e dx+A, l-n (\-ri)fPdx ,., v f^ (l-ri)) i.e. y e =(1 ri) We Ex.1, Integrate -^ + ^=?/ 2 . Here ^-2^+^ = 1; or putting -=0, t7 x-, A dx+A. dx x and the integrating factor being -fix* e j * = e - loex we have ^(^=-1, dx\x) x i.e. ?=logi X X i.e. -=Ax y Ex. 2. Integrate the equation -jf. + x sin 2y = Cv^7 Dividing by cos 2 y we have ec 2 y-^ + 2# tan yx^. dx Putting we have tany=2, ^ + 2^=^, GW? and the integrating factor is J 2xdx O r e* z , giving LINEAR EQUATIONS. 219 Let x 2 = co, then 2# dx = d(v)> dv _dx ~(p) ............................... (1) Differentiating with respect to x, or dx = <[>'(p)dp x -< Integrating this equation we have x expressed as a function of p and an arbitrary constant Ax=F(p)(**y) ......................... (2) Eliminating p between equations (1) and (2) we obtain the solution required. Ex. 1. Solve (x*+y*)ty-=xy. dx and putting y=v%, ^+v dx dv or x = - dx or og=- 2 a;2 or Ay^eW. Ex. 2. Suppose the equation to be x dx \dx) ' HOMOGENEO US EQ UA TIONS. 223 Then p = (p +p 2 ) + x(l + 2p), p giving log J,#+2logp -=0, P i.e. and the jo-eliminant between p 2 +p= x \ and Axp*=& < is the solution sought. This eliminant is But when it is algebraically impossible to perform the elimination of p, or when, if performed, the result will be manifestly unwieldy, it is customary to leave the two equations containing p unaltered, and to regard them as simultaneous equations whose jo-eliminant if found would be the required solution. EXAMPLES. Solve the differential equations . .=. dx x+y 2. 224 DIFFERENTIAL EQUATIONS. 179. A Special Case. The equation ~ f - ^- - , is readily reduced to dx ax+oy+c the homogeneous form thus : Put x TVi ^ a ^+ by + (ah + bk + c) dg- a 'g + b'' ' Now choose h, k so that .1 ^e. so that r - / ^- = - ,- -^ - ^ oc be ca ca ao a 6 Then ^ = This equation being homogeneous we may now put n~ v ^ an( i ^ ne variables are separable as before shown. 180. There is one case, however, in which h, k cannot be chosen as above, viz., when a _ b c a' ~~ 6' c'' Now let =m and a dy Then Tx= so that -~ a = n 7 -- * - V - - /) dx my + c drj _ (am + b)rj + ad + 6c dec"" mrj + c and -, - , , >. . -, n. (am-\-o)r)-\-ac +bc HOMOGENEOUS EQUATIONS. 225 The variables being now separated, the integration may be at once performed. 181. One other case is worthy of notice, viz., dy _ ax + by + c dx~ bx + b'y + c" when the coefficient of y in the numerator is equal to that of x in the denominator with the opposite sign. For then we may write the equation thus (ax + c)dx + b(y dx+x dy) = (b'y + c')dy an " exact " differential equation ; the integral being ax 2 + 2cx + 2bxy = b'y 2 + 2c'y + A , A. being the arbitrary constant. Ex.l. Integrate = y- dx x+y-Z Put #=+ k, y = ri + k, so that Choose h and Ic so that = then Now put 77=0(1, then _ l+v ~ v+1 ' - 1) 2 - l where =#1 and v=^~ . x\ E. I. c. p 226 DIFFERENTIAL EQUATIONS. Ex. 2. Integrate f = *+* . dx x+y \ Let .#+y=??, then .. . = dx ?? - 1 if] 1 ' and ^ where ?7= EXAMPLES. Integrate the equations : dy _ dx bx+ay-b ^ 1* 8 9. Show that a particle #, y which moves so that ~ will always lie upon a conic section. 10. Show that solutions of the general homogeneous equa- tion fUL ~) must always represent families of similar ' \ dx) curves. 11. Show that solutions of /(-, -j-} are homogeneous in x, \ X CLX J y and some power of a single constant, and conversely that if the typical equation of a member of a family of curves be homo- geneous in x, y and some power of one constant, the differential ONE LETTER ABSENT. 227 equation of the family is homogeneous and the family consists of similar curves. 12. State which of the following families of curves are similar sets : (1) ya = 4flW7. (4) 2/ = (2) y = a cosh -. (5) b tan" 1 * 7 = a +y. for different values of a and b. 182. CASE IV. One letter absent. x absent. A. Suppose x absent from the differential equation, which then takes the form Xdy\ - y> ZH we now solve for ~ or y, as may be most convenient. (i.) If we solve for -^ we throw the equation into the form Then and the integral is (ii.) If this be inconvenient or impossible we may solve for y and obtain y = ( j ) (p) ) where p stands as before for --. ax 228 DIFFERENTIAL EQUATIONS. Differentiate with regard to x, i.e. the absent letter. The P and dx Thus x After the integration is performed we eliminate p between this equation and 2/ = (p) and the solution of the given equation is obtained. 183. y absent. B. Suppose y absent from the differential equation, which then takes the form fj 1J Since -^ = this may be written ax ax dy dx' and therefore if y be regarded as the independent variable the foregoing remarks apply to this case also. Thus dx (i.) if convenient we solve for -^-, and obtain a result of the form ^ dx , . a5T*S 7 dx then dy = 7-^, and the integral is dx ONE LETTER ABSENT. 229 (ii.) But if this solution for -7- be inconvenient or dy impossible we solve for x and obtain a result of the yy /v> form x = (j)(q) where q stands for -j- Then differen- tiating with regard to y, the absent letter , Thus and After the integration we eliminate q between this equation and x = -eliminant is the solution in question but the actual elimination not performed. EXAMPLES. Solve the equations : 1. y= , 2. y = 3. y= 4. y= p 234 DIFFERENTIAL EQUATIONS. 8. The tangent at any point P of a curve meets the axis Oy in T, and OT 2 is proportional to the tangent of the inclination of PTto the axis Ox. Find the curve. [OXFOKD, 1888.] 9. Find the differential equation of all curves which possess the property that the sum of the intercepts made by the tangent on the coordinate axes is constant. Obtain as the complete primitive the equation of the tangent, and as the singular solu- tion the curves in question. 10. Obtain the curves for which the area of the triangle bounded by the axes and a tangent is constant. 11. Form the differential equation of curves for which the, length of the portion of the tangent intercepted between the coordinate axes is constant. Obtain and interpret the complete primitive and the singular solution. 12. A curve satisfies the differential equation y=p\xp)) and also that^>=0 when x=\ ; determine its equation. [OXFOKD, 1889.] IS. Find the complete primitive and singular solution of the equation dx \ V^yj ' [OXFORD, 1890.] 14. Show that by putting x 2 s and y 2 = , the equation is reduced to one of Clairaut's form. Hence write down its complete primitive and find its singular solution. Interpret the result. CHAPTEE XV. DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. EXACT DIFFERENTIAL EQUATIONS. 187. Second Order Equation. We next come to the consideration of the differential equation of the second order, 0(^2/>2/i>2/2) = - There is no general method of solution, but particular forms arise which present but little difficulty. 188. CASE I. Suppose the Equation linear. The typical form will be dx where P, Q, R are functions of x. The usual method is first to omit R and try to obtain or guess a solution of Suppose y =f(x) to be such a solution. Put y =2/0*0- 236 DIFFERENTIAL EQUATIONS. Then yi = zj(x)+zf(x); 2/2 = zj(x)+^'(x)+zf"(x). Thus on substitution we get But /'(*) + Pf(x) + Qf(x) = by hypothesis. Hence z an equation which is linear for z v The integrating factor is or and the first integral is whence the second integral may be at once obtained and the solution effected. Ex. Solve L da? dx Here y=x makes -r^ Put then Hence /K 4- 3^ ** and the integrating factor is e^ x or x*e 4 . SECOND ORDER EQUATIONS. 237 j x * Thus ~(z l x^)=x^ a* 5 and z 1 x 2 e*=~+A _* r , whence z=-\e * + A \ -e J a? and the solution required is -= 5 189. CASE II One letter absent. A. If x be absent, let y 1 =p, - and the equation (y, y v y 2 ) = takes the form (y, p, p-S-\ = Q, \ dy/ and is of the first order. B. If y be the letter absent, let y l =p, *- and (x, y v y%) becomes and again is of the first order. Ex.1. Solve the equation yy 2 +#i 2 =2# 2 . Here x is absent. So putting y=p and y 2 =p^?, we have The integrating factor is e^ v Ay or y 2 , or p 2 y 2 y* + constant =?/ 4 + a 4 , say. 238 DIFFERENTIAL EQUATIONS. Hence or i. e. y* = a 2 sinh(2# + A). Ex. 2. Solve l+#i 2 =#y 2 #i Here y is absent. So putting y-^p^ dx pdp or = ---5, x 1+jtr i. e. log x = log Vl+p 2 + constant. ^.e. or ady ^Jx* a? dx, giving oy =i?^ 2 _< a and b being arbitrary constants. EXAMPLES. Solve the following equations : 1. ^ 2 = 1. 6. 2. 1+3^=^2. 7. 1 _L 2 3. i+y! 2 -^ 2 - 8 - y2+/i-y=-e 2 - ^ having given that ^ = when y = 0. [OXFORD, 1890. ] ow? 11. Given that # 2 is a value of y which satisfies the equation find the complete solution. [L 0. S., 1894.] REMOVAL OF A TERM. 239 190. General Linear Equation. Removal of a Term. Let us next consider the more general equation where P v P 2 , . . . , Q are given functions of x. Putting y = vz, we have y 2 = vz 2 + 2V& + v 2 z, etc., whence , n(n 1) - -- - 2 The coefficient of n _i is If then v be chosen so that dv P or v = e v n the term involving z n -i will have been removed. Similarly, if v be so chosen as to satisfy the differ- ential equation the term containing W _ 2 will have been removed. The coefficient of z is and if a value of v can be found or guessed which will make this expression vanish, we can, by writing z l = rj, and therefore z 2 = rj ly etc., and z n = rj n -i, reduce the degree of the equation by unity. The student should notice that this expression is the same in 240 DIFFERENTIAL EQUATIONS. form as the left hand member of the given equation. Hence if any solution y v can be found or guessed of the given equation when the right hand member is omitted, we can, by writing y = vz, and then Z^ Y\, reduce the degree of the equation. 191. Canonical Form. In the case of the equation of the second degree the substitution y = e~ l ^ dx z will by what has been above stated reduce the given equation to the sometimes simpler form But the general solution of this equation has not been at present effected. "EXACT" DIFFERENTIAL EQUATION. 192. When p is < q. x p -r~ is an exact differential. ^ and can be integrated whatever y may be. For denoting by y q , \xPy q dx = etc., Thus EXACT DIFFERENTIAL EQUATION. 241 It will be noticed that when q =p or < p the in- tegration cannot be effected. 193. By aid of the above lemma we may often see quickly whether a given equation is " exact/' For if all terms of the form x p y q in which p is < q be first removed, we can frequently tell at once by in- spection whether the remainder is a perfect differential coefficient or not. Ex. # 2 Here, by the lemma, # 2 y 5 and x?y are perfect differential coefficients, and obviously ocy^-\-y is the differential coefficient of xy. Hence a first integral of this differential equation is obviously =- cos x+A. 194. A more General Test. A more general test for an "exact" differential equation may be established in the general case . whatever forms the coefficients P , P v ... , P n , V may have, provided they be functions of x. For denoting differentiations by dashes, we have upon integration by parts n- Bysdx =P n - 32/2 - Pn - 8^1 + P"n - zV - I P"'n - etc. Hence upon addition it is obvious that if p. i. c. 242 DIFFERENTIAL EQUATIONS. the given equation is exact; and that its first in- tegral is Ex. Is the equation cfiy z + 1 2x?y 2 + SG^ 2 ^ + 24#?/ = sin x exact ? Applying the test, we have and P 3 - P 2 ' + PI - PQ'" = 24^ - 72^ + 72^ - 24^ = 0, Thus the equation is exact ; and its first integral is or This again will be a perfect differential if which is satisfied. Hence a second integral will be (8# 3 - 43% -f ^ 4 y x = - sin x + Ax + B, or 4^7 3 y+^ 4 y 1 = sin^+^^ + J?, which may again be tested. But it is now obvious that the third and final integral is ^ = cos.*?+ IB EXAMPLES. 1. Show that the equation exact, and solve it completely. 2. Solve the equation . ^7/3 + 6^/2 + %i + sin x (y* ~ %i) + cos X 3 3/2 - !/} = sin ^ 3. Write down first integrals of the following equations : '(a) (b) (c) 4. Show that if the equation P 2 y + P^y^ + P 0< y 2 = F admits of an integrating factor //,, then //, will satisfy the differential equation CHAPTER XVI. GENERAL LINEAR DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS. 195. General Linear Differential Equation. The form of. the general linear differential equation of the nih order is where P v P 2 , P 3 , ..., V are known functions of x. Let us suppose that any particular solution, y=f(x) can be guessed, or obtained in any manner. Then making the substitution ~ we obtain Suppose z = z v z = z 2 , . . . , z = z n to be solutions of this equation ; then it is plain that z = A l z l + A 2 z 2 +A 3 z 3 +... +A n z n is also a solution of equation (2) containing n arbitrary constants A lt A 2 , ..., A n . 244 DIFFERENTIAL EQUATIONS. Hence is a solution of equation (1) containing n arbitrary constants, and is therefore the most general solution to be expected. No more general solution has been found. The portion f(x) is termed the Particular Integral (P.I.), and the remaining part containing the n arbitrary constants, which is the solution when the right-hand member of the equation is replaced by zero, is called the Complementary Function (C.F.). If these two parts can be found the whole solution can be at once written down as their sum. 196. Two remarkable Cases. There are two cases in which these solutions can be generally readily obtained. (1) When the quantities P v P 2 , ..., P n are all constants. (2) When the equation takes the form ri Jn-l r 7n-2n/ 2+ -'- + ^= F > ' a v a 2 , ..., a n being constants and V any function of x. The solution of the second case is readily reducible, as will be shown, to the solution of an equation coming under the first head. EQUATION WITH CONSTANT COEFFICIENTS COMPLE- MENTARY FUNCTION. 197. Let us therefore first determine the solution of such an equation as 2/n+^i2/u-i + a 2 2/n- 2 +... + ^n2/ = 0, ......... (1) the coefficients being constants ; i.e. for the present we confine our attention to the determination of the " Complementary Function " in the first case. COMPLEMENTAR Y FUNCTION. 245 As a trial solution put y = Ae mx , and we have m n + a 1 m n - 1 + a 2 ra w - 2 +...+a n = ....... (2) Let the roots of this equation be 774, m 2 , m s , ..., m n , supposed (for the present) all different, then are all solutions, and therefore also y = A^ x + A 2 e^ x + A 3 e m * x +...+A n e m * x , ...... (3) is a solution containing n arbitrary constants A v A 2 , A 3 , ..., A n , and is the most general to be expected. 198. Two Roots Equal. If two roots of equation (2) become equal, say m x = m 2 , the first two terms of the solution (3) become (A!+ J. 2 )e" 11 *, and since A^ + A^ may be regarded as a single constant, there is an apparent diminution by unity in the number of arbitrary constants, so that (3) is no longer the most general solution to be expected. Let us examine this more closely. Put 97i 2 = m 1 +A. Then A ^ x + A 2 e( TO i+*X r h?x 2 ~~\ = A l e m ^ x -\-A 2 (^ x \ l+hx+-^- + ... rhy? ~\ = (A l + A 2 )^ x + AJi . xe m ^ x +A z he m ^~^ + . .. I. Now A^ and A 2 are two independent arbitrary quantities, and we may therefore express them in terms of two other independent arbitrary quantities by two relations chosen at our pleasure. First we will choose A 2 so large that ultimately A 2 h when h is indefinitely small may be written 2 , an arbitrary finite constant. 246 DIFFERENTIAL EQ UA TIONS. Secondly, we will choose A 1 so large and of opposite sign to A 2 that A^+A 2 may be regarded as an arbitrary finite constant B v Then the terms ultimately vanish with h since Aji has been considered finite and the expression in square brackets is con- vergent and contains h as a factor. Thus the terms A^^+A^e 11 ^ may, when m 2 = m v be ultimately replaced by B 1 e miX +B 2 xe miX > and there- fore the number of arbitrary constants in the whole solution remains n, and we therefore have obtained the general solution in this case. 199. Three Equal Roots. Consider next the case when three of the roots of equation (2) become equal, viz., m 1 = m 2 = m 3 . The terms, A l e m &+A# m * x +A 2 f? ri * x , have already been re- placed by (Bi+B^e^+Atf**. Let m s = m x + h / fcZftZ \ Then A^ x = AjPtifc = A^ x ( 1 + kx + -^- +...) Thus for A+Aw + Ae'W we have and we may so choose A B , 5 2 , and B v that O v C 2 , 3 being any arbitrary constants, whatever k COMPLEMENTARY FUNCTION. 247 may be, provided it be not absolute zero. But AJc 2 being chosen a finite quantity, and the series within the square brackets being convergent, it is clear that ultimately, when k is indefinitely diminished, the limiting form of this expression is 200. Several Roots Equal. In a similar manner it will be obvious that if p roots of the equation (2) become equal, viz., m^ = m 2 = . . . = m p , there will be no loss of generality in our solution if we substitute the expression (%! + K^x + Kfl? + . . . + KjxP - *)&*&, for the corresponding portion of the complementary function, viz., A^ x + A 2 e m * x +...+ A p e w *> x . 201. Generalization. More generally, if be the complementary function of any linear differ- ential equation with or without constant coefficients, what is to replace this expression so as to re-tain the generality when m x = m 2 ? Let m 2 Then and the terms A l (m 1 ) + A 2 (m p ) may be replaced by when the generality of the solution will be retained. The results of Arts. 198, 199, 200 are of course par- ticular cases of this, the form of 6 a *sin bx sm bx, COMPLEMENTARY FUNCTION. 249 where the two arbitrary constants B l and B 2 replace A^+ A 2 and (A l A 2 )i respectively. Let B^ = p cos a, B% = p sin a, then = JB* + 2 2 and a = tan - ijgF. Then ^cos fr# + 2 sin bx = p cos(bx a). We may thus further replace jB^cos bx + B 2 e ax sin bx by C L e aa! cos(6aj where C^ and 2 are arbitrary constants. 203. Repeated Imaginary Roots. For repeated imaginary roots we may proceed as before, for it has been shown that when 777 2 = ??i 1 , A l e m ^ x +A^ x may be replaced by (J^+J?^***, and if m 4 = m 3 , A^ X +A^ X may be replaced by If then m x = m 2 = a + ib and m 3 m 4 = a f 6, we may replace by ( that is by e ax [(B l + 5 3 )cos bx + (B l - B 3 )i sin te] + xe ax [(B 2 + ^ 4 )cos 60? + (B 2 - 4 )^ sin 6 and therefore by e^CC^cos 6^+ (7 2 sin 6aj) +cce aaj ((7 3 cos 6x+ (7 4 sin that is by tf*(Ci + cc(7 3 )cos 6aj + ^ or which is the same thing by Any of the last three forms contain four arbitrary constants which replace the original four arbitrary constants A v A 2 , A^ A^ and thus retain intact the 250 DIFFERENTIAL EQUATIONS. proper number (n) of arbitrary constants requisite to make the whole solution the most general to be expected. And this rule may obviously be extended to the case when any number of the imaginary roots are equal. 204. Ex. 1. Solve the equation ^- dx 2 dx Here our trial solution is y=Ae mx y and we obtain whose roots are 1 and 2. Accordingly y A^e* and y = A 2 e 2x are both particular solutions, and y=A 1 e*+A 2 e 2x is the general solution containing two arbitrary constants. Ex.2. Solve -*V- a?y =0. aOC Here the auxiliary equation is w 2 a 2 = with roots m a, and the general solution is or as it may be written (if desired) y = .Z^cosh ax + ^sinh ax by replacing A l by B i +B * and A 2 by B ^~ B 2 2 Ex.3. Solve Here the auxiliary equation is m 2 -f-a 2 =0 with roots m +ai. Hence the general solution is y = AjCos ax + ^t 2 s i or, which is its equivalent, y = B l Ex.4 Solve ?-4| + 5-2y = or (D- l) 2 (-2)y = 0, ax? ax ax where D stands for . ax ILLUSTRA TIVE EXAMPLES. 251 Our auxiliary equation is or m-lra-2 = 0, having roots 1, 1, 2. Accordingly the general solution is Ex. 5. Solve (Z> 2 +l)(7)-l)y=0. Our auxiliary equation is with roots i, 1, and the general solution is therefore or ?/ = ^008(37 + J5 2 )-h Atf?, Ex. 6. Solve Our auxiliary equation has roots ^ e^L. and 2, and the general solution is JQ cos + ^I 2 e sn 2 2 Ex. 7. Solve (Z) 2 + />+l) 2 (Z>-2) 3 (/)-5)y=0. Here obviously the general solution is y = (A l + A 2 x)e ^cos x ^ + ( A 3 + ^t 4 #)e~^sin + (A 5 + A Q x + J.7^ 2 ; containing eight arbitrary constants. EXAMPLES. Write down the solutions of the following differential equa- tions : 252 DIFFERENTIAL EQUATIONS. 3 - S- 9 4- S~ 3 5. g=y. 9. 6. g=y. 10. 11. (- 12. THE PARTICULAR INTEGRAL. 205. Having considered the complementary function of such an equation as F(D)y = V where F(D) stands for a v a 2 , ..., a n being constants, and Fany function of x, we next turn our attention to the mode of obtaining a particular integral, and propose to give the ordinary and most useful of the processes adopted. We may write the above equation as 2/ = 1 (or [/(D)]F), where ^7^ is such an operator that 206. "Z>" satisfies the fundamental laws of Algebra. It is shown in the Differential Calculus that the operator D (denoting -y- j satisfies (1) The Distributive Law of Algebra, viz. (2) The Commutative Law as far as regards con- stants, i.e. D(cu) = c(Du}. PARTICULAR INTEGRAL. 253 (3) The Index Law, i.e. m and n being positive integers. Thus the symbol D satisfies all the elementary rules of combination of algebraical quantities with the exception that it is not commutative with regard to variables. It therefore follows that any rational algebraical identity has a corresponding symbolical operative analogue. Thus since by the binomial theorem 7? ( ' T) " ~L i l + \ JL . 2i we have by an analogous theorem for operators which may be inferred without further proof JL . < 207. Operation f(D)e ax . It has been proved in the Differential Calculus that if r be a positive integer, Let us define the operation D~ r to be such that D r D~ r u = u. Then D~ l represents an integration, and we shall suppose that in the operation D~ l u no arbitrary con- stants are added (for our object now is to obtain a particular integral and not the most general integral). Now since D r a~ r e ax = e ax = D r D- r e ax , it follows that D~ r e ax = a- r e ax . Hence it is clear that D n e ax = a n e ax for all integral values of n positive or negative. 254 DIFFERENTIAL EQUATIONS. 208. Let f(z) be any function of z capable of ex- pansion in integral powers of z, positive or negative ( = ^A r z r say, A r being a constant, independent of z). Then The result of the operation f(D)e ax may therefore be obtained by replaci'ng D by a. Ex. 1. Obtain the value of - ^ -e Obviously by the rule this is "* or g. E, 2 , Obtain the value of By the rule this is e 3a/ '=-^-e EXAMPLES. 1. Perform the operations indicated by ' ^ 3. Apply Art. 208 to show that /(Z) 2 )sin m^7 =/( m 2 )si /(Z) 2 )cos mx =/( m 2 )c 209. Operation f(D)e ax X. Next let y = e ax Y, where Fis any function of x. PARTICULAR INTEGRAL. 255 Then since D r e ax = a r e ax , we have by Leibnitz's Theorem y n = e dx (a n F+ n (7 1 a n - 1 D Y+ n C 2 D 2 Y+...+D n F), which, by analogy with the Binomial Theorem (Art, 206), may be written D n e ax Y= e ax (D + a) n F, n being a positive integer. Now let X so that we may write Then from above D n e ax Y=e ax (D+a) n Y or D n e ax (D + a) ~ n X = e ax X, and therefore D Hence in all cases for integral values of n positive or negative D n e ax X = e ax (D + a) n X. 210. As in Art. 208 we shall have f(D)e ax X = That is, e ax may be transferred from the right side to the left of the operator f(D) provided we replace D by D + a. Ex. 2. - - - e 2x sin x = e 2x ~ sin x e~ x sin x. D 2 4D + 4 D * 256 DIFFERENTIAL EQUATIONS. EXAMPLES. 1. Perform the operations 1 o 1 1 (D - If* X ' (D-I)* 6 h l X ' D-l 2. Show that 211. Operation /(7> 2 ) We have D 2 sn wwu = ( - m 2 ) sin mx, cos y cos and therefore Hence, as before, Arts. 208 and 210, it will follow a ^ j;/ r>9\ sin /v o\ sin n D ) mx = f( m 2 ) mx. J ^ ' cos y cos Ex. f e ax sin bx dx = Z)- 1 e ax sin 6^7 = e ax (D + a)~ 1 sin 6^ (Art. 210) 7)Wri hv f'Arf 91^ X/^iSJlll f)cos mx, /(Z>)sin mx. PARTICULAR INTEGRAL. 257 212. Operation Let us next consider the operation where F(z) is a function of z capable of expansion in positive integral powers of z. Let F(D) be arranged in powers of Z), then if no odd powers occur the result may be written down by the foregoing rule of Art. 211. Thus "" S1T1 ^ = L-4 + 16-64 " ~ 51 sin 2# = sin %x sin 2#. But if both even and odd powers occur we may proceed as follows : Group the even powers together and the odd powers together, and then we may write the operation sm mx = . /T ^ox . ^ /TV,V sin mx ^ ^ m 2 )sin mx m x ( m 2 )cos mx Upon examination it will be seen that in practice we may write m 2 for Z) 2 immediately after the step writing immediately 1 \ f)~7 ^ sin mx> E. I. C. R 258 DIFFERENTIAL EQUATIONS. mi in/ ) J-'Xv i in i . , or' r-y^ SYT^ fjsrS 4^0 sin ma;, etc. Ex. 1. Obtain the value of ^ = - sin 2#. Thisis sin 2#, D sin 2^, - J.O or ^ cos 2^ ^ sin 2^7. Ex. 2. Obtain the value of ^ -^e 2 *cos ^p. This expression = e 2 *-^ - cos x ^ [replacing each Z> 2 by 1] e 2 * 1 ____ = (cos x sin x). 4 EXAMPLES. 1. Perform the operations indicated in the following ex- pressions :- D Z> 3 -e*sin x + PARTICULAR INTEGRAL. 259 2. Show that /T , l V = e~ ax f [ f ... fe ax Vdx ... dx, there (D+a) n J J J J being n integral signs. 3. Show that by first expressing ^=-r~ in partial fractions, the 1 *W operation ^fm F may be expressed in terms of a set of common integrations. 213. Operator rV- v Algebraic. If in the operation wy^F, Fbe an algebraic function of x, rational and integral, we may expand rrrr\\ by any method in ascending powers of J9 as far as the highest power of x contained in F. Ex. 1. For example, find - This is l (1 -D+ D 3 - D^+ etc.)(# 2 Ex.2. Again, find This expression is 10 260 DIFFERENTIAL EQUATIONS. EXAMPLES. Perform the operations CD+l)(Z> + 2) 2 - 3. - J? COsh 37 COS 07. 214. Cases of Failure. In applying the above methods of obtaining a Particular Integral, cases of failure are frequently met with. We propose to illustrate the course of procedure to be adopted in such cases. 215. Ex. 1. Solve the equation ( ^L-y=e x . dx ' The Complementary Function is Ae*. To obtain the Particular Integral we have If we apply Art. 208, the result becomes i^i or - We may evade this difficulty and obtain the result of the operation by applying Art. 210 when we have which is the particular integral required. Instead, however, of substituting another method, let us examine the operation = -& more carefully. Writing x(\ + h) instead of #, we have 263 CASES OF FAILURE. Of this expression the portion Lte x //i becomes infinite, may be taken with the complementary function Ae x ; and A b arbitrary we may regard A + - as a new arbitrary constan /i for we may suppose A to contain a negatively infinite por to cancel the term I/A. The term xe* is the Particular Integral desired. The remaining terms contain h and vanish when h is decre; indefinitely. The whole solution is Ex.2. Solve the equation ^ Ct/X The complementary function is clearly y = A sin Zx + B cos 2#. The particular integral consists of two parts i e* o and sin 2#. In this second part, if we apply the ri Art. 211, we get 2HL, i.e. oo , and so fail. We now consider the limit, when h = Q, of -- sin2.i'(l- This expression = 1 _1 4 i_(i+A)a 1 1 I 9A JT2^ U ^ X COS < ^ IX "*" COS ^^ S ^ n ^^ J? ) - 1 sin 2.2? 1 r , l -x cos 2.27 + powers of A o fi 4 = (a term which may be included in the complem- function) - x c ^ s ^ + (terms which vanish v Thus the whole solution of the. differential equation g JOS " 260 DIFFERENTIAL EQUATIONS. :. 3. Solve the equation (D 2 + 3D)(D -l) 2 y = e* + e~ x + sin x 4- x 1 . 'ere the complementary function is plainly particular integral consists of four parts, viz., 1 x _ 1 . in #*, l . plX * rp -4^-6/^Tr. ' ** l l ~^l-^D..\^ Thus the P.I. is COSJP _ 3^ gin ^ 8 8 and the whole solution is y = A ^inh ^ + ^ 2 cosh x + A 3 sm x+A 4 cos x + x c ^ s ^' - ^ sin ^p. EXAMPLES. 1. Obtain the Particular Integrals indicated by 0) 7TTT sina7 ' ( 5 > /n w^ovn-Q-x*'- (6) _ (sinh # + sin ^?). nT-T 8 * 1111 *' ( 7 ) /na ^/ n ^o,(^ + cosh 6^). COS - COS . o o 264 DIFFERENTIAL EQUATIONS. 2. Solve the differential equations (3) +y = Cfc# (4) (D*-l)(D*-l)y=xe*. (5) (Z)- (6) (ZP-3D 2 -3D + I)y = e- x (7) OD 3 -%=#sin.. (8) (Z> 2 (9) (Z> 2 (10) (^>- 216. The Operator &-. CvQC A transformation which renders peculiar service in reducing an equation of the class where A v A 2 , ..., are constants, to a form in which all the coefficients are constants, arises from putting x = e t . In this case -TT = e*, and therefore x~- = -^- at ax at It is obvious therefore that the operators x-j- and d d dx -ji are equivalent. Let D stand for -j-. Then we have dx\ -*- n )X n _ . 1 nf\ n _ Z. ( x __ 11 -I- 1 ]X ~ ___ dx n \ dx / dx n ~ l EXAMPLES. 265 Now putting 11 in succession 2, 3, 4, ..., we have , etc. Hence generally or reversing the order of the operations = D(D-l\D-Z)...(D- Ex. Solve the differential equation Putting x = c?, the equation becomes D(D -l)(D- 2)y + 2D(D - % + 3% - 3# = or ( i.e. (D - giving y = Ae* + B cos - , ~+ ;rIog EXAMPLES. Solve the differential equations 1. s 2 dx 2. x- --^ + a? -^ + (^ = [log ^-] 2 + x sin log ^ + sin q log a?. 3. + 3^++2/=^ cfc 1 ote 2 rfa? ' 4. i dx? dx* dx 5. CHAPTER XVII. ORTHOGONAL TEAJECTOEIES. MISCELLANEOUS EQUATIONS. ORTHOGONAL TRAJECTORY. 217. Cartesians. The equation f(x, y, a) = is representative of a family of curves. The problem we now propose to investigate is that of finding the equation of another family of curves each member of which cuts each member of the former family at right angles. And in such a problem as this it has been already pointed out that it is necessary to treat all members of the first family collectively, so that the particularizing constant a ought not to appear in the equation of the family. It has been shown in Art. 17 1, that the quantity a may be eliminated between the equations . *dx 'dy dx Let this eliminant be This is the differential equation of the first family. ORTHOGONAL TRAJECTORY. 267 Now at any point of intersection of a member of the first system with a member of the second system, the tangents to the two curves are at right angles. Thus if r\ be the current coordinates of a point on a curve of the second family at its intersection with one of the first family, and x, y the current co- ordinates of the same point regarded as lying upon the intersected curve of the first family, we have * dn dx t-x.i-y.g^ The differential equation of the second family is therefore and when this is integrated we have the family of " Orthogonal Trajectories " of the first system. The rule is therefore : Differentiate the given equation, eliminate the rtnt* fl 1J constant, write -7- in place of -j , and integrate the new differential equation. 218. Polars. If the curve be given in polars the angle the tangent df) makes with the radius vector is r-j-, so our rule is now: dr Differentiate the equation, eliminate the constant, ., 1 dr . , ite -70 in plac differential equation. orthogonal trajectory of the family (1) ., 1 dr . , dO , . . write -70 in place of r-v-, and integrate the new 219. Ex. 1. Find the orthogonal trajectory of the family of circles each of which touches the y-axis at the origin. 268 DIFFERENTIAL EQUATIONS. Here x+yJL= a , cLx and, eliminating #, .v 2 +y 2 = 2x( x -\-y ), Hence the new differential equation must be or ^ 2 + 2^-^2 = o, ........................... (3) ay which is a homogeneous equation, and the variables become separable by the assumption y = vx. However, this being the same as equation (2) with the ex- ception that x and y are interchanged, its integral must be another set of circles, each of which touches the #-axis at the origin. Ex. 2. Find the orthogonal trajectory of the curves 2 i n\ -- A being the parameter of the family. and A must be eliminated between these two equations. (2) gives x(b* + A) +yy l (a? + A) = 0, so that a 2 + A = and Thus the differential equation of the family is (at-b^yy or x*-y*+xyyi- =a 2 -5 2 ................ (3) ORTHOGONAL TRAJECTORY. 269 Hence changing y^ into , the differential equation of the #1 familv of trajectories is 2 (4) But this being the same as equation (3) must have the same primitive, viz. : n a ^ i y 2 _-. * ~ i.e. a set of conic sections confocal with the former set. Ex. 3. Find the orthogonal trajectories of the family of cardioides r=a(l cos 0) for different values of a. Here ^ and, eliminating a, r^ = l ~ = = dr sm 2 Hence for the family of orthogonal trajectories we must have 1 dr n or log r 2 log cos + constant, 2t or r=b(l+cosO), another family of coaxial cardioides whose cusps point in the opposite direction. EXAMPLES. 1. Find the orthogonal trajectories of the family of parabolas # 2 = 4cM? for different values of a. 2. Show that the orthogonal trajectories of the family of similar ellipses - 4-^,=m 2 for different values of m is s? =Ay b . a 2 b 2 3. Find the orthogonal trajectories of the equiangular spirals r = ae^ cota ' for different values of a. 4. Find the orthogonal trajectories of the confocal and coaxial parabolas =1 -f cos 9 for different values of a. 270 DIFFERENTIAL EQUATIONS. 5. Show that the families of curves are orthogonal. 6. Show that the curves r sin 2 a = a(cos cos a) and r sinh 2 /? = a(cosh ft cos 0) are orthogonal. 7. Show that if f(x+iy) = u + iv the curves form orthogonal systems. 8. Prove that for any constant value of /z the family of curves cosh x cosec y p cot y = constant cut the family /z coth x cosech x cos y = constant at right angles. [LONDON, 1890.] SOME IMPORTANT DYNAMICAL EQUATIONS. 220. The equation is the general form of the equation of motion of a particle under the action of a central force. Multiplying by 2-^ and integrating we have dO which we may write as and the solution is therefore effected. 221. Equations of the form SOME SPECIAL FORMS. 271 have already been discussed as being linear with con- stant coefficients. The solution may however be conducted thus : Multiply by sin n9, which will be found to be an integrating factor. Integrating, sin nO^. - nu cos nO= f*f(ff) sin n&dff + A. d6 J o Similarly, cos nO is an integrating factor and the correspond- ing first integral is cos n(& + nu sin nO= f'f(ff) cos nO'dO'+B. d\j J o Eliminating -^L du nu = e f(0') sin n(0 - O')d0 f + Bsmn6-A cos n6. 222. The equation of motion of a body of changing mass often takes some such form as d! dt ^ and for this equation (x)-rr will be found to be an integrating factor. leads at once to i{<(^' J 2 = j ^x)(x)dx + A, 1 ^Xto J and the variables are separated. 272 DIFFERENTIAL EQUATIONS. FURTHER ILLUSTRATIVE EXAMPLES. 223. Many equations may be solved by reducing to one or other of the known forms already discussed by special artifices. Ex. 1. ^=f(ax+by). ax ' Let ax+by=z. Then = dx dx Thus dx dz or x+A=l d *.. J a + bf(z) Ex.2. y+a dx\ dx Put xy=z. rpi . dy dz Then y+^^=-y-, dx dx dz , 1 Z = X-j- +-5- or dx dz dx which is of Clairaut's form, and the complete primitive is . Ex.3. Solve e^ - dx) \dx Let 6^ = 77, e x = s- Then, since this equation may he arranged as dx \ e *dx ILL USTRA Tl VE SOL UTIONS. 273 we may write it as ?7 which being of Clairaut's form the complete primitive may be written or Ex. 4. -- (an. equation occurring in Solid Geometry). Put ,v=*Js and y = Jt. Then the equation becomes , ds giving t= ds as which is of Clairaut's form and has the complete primitive t-sG BC ~ 1+2(7' and singular solution the four straight lines 9J-Jy Ex. 5, Solve the equation dx E. I. C. S 274 DIFFERENTIAL EQUATIONS. Let the transformation be such that then x is known by direct integration as a function of t. dy Now d^d L _ dx and * ' dx* Thus (^ax^yJ^- ax d 4 . f , } dx* dt* dt dx and the given equation thus reduces to whose solution is y=A sin qt + B cos <^, and when the value of t in terms of x is substituted, the solution is complete. [If a be positive we have 1 dx ,. -[=. >- ~j= sinh^Wa) = t. If a be negative we have 1 dx V-a ,=dt> Ex. 6. Solve the simultaneous differential equations (which are linear with constant coefficients) SIMULTANEO US EQ UA TIONS. 275 We may write these equations as (3 D + 34>? + (ID + 38)y = e\ where D stands for . at Operating upon these equations respectively by 7J9 + 38 and by 9D + 49 and subtracting, we eliminate y and obtain [(4Z) + 44X7/> + 38) - (3D + 34)(9Z> + 49)> = 7 + 38* - 58e, or (D 2 + 7Z> + 6)ff = 7 + 38* - 58e, giving ^= or a? = ^e To obtain y let us eliminate -^ from the original equations. Multiply the first by 7 and the second by 9 and subtract. This gives ^ + 1x +y = It - 9e*. at Thus y = 7^-9^-2^-^ = It - 9e* - 2( Ae~* + Be ~ Thus ^=^-* + ^- 6 + - 1 3 9 -^--^--^V ? \ .y = - Ae-< + 4^e - 6 ' - *t + * + *fr*. ) [The student should notice the elimination of ^-. This avoids the introduction of supernumerary constants.] at Ex. 7. Solve the simultaneous equations ^+ 3 f dt 2 dt dt These equations may be written 276 DIFFERENTIAL EQUATIONS. whence operating upon these in turn by D 2 + 9 and by 3D and subtracting, we eliminate y and obtain [(D 2 + 16)(D 2 + 9) + 1 5 Z) 2 > = 0, or (Z> 4 + 40Z) 2 + 1 44> = 0, i.e. (D* + 4)(D' 2 + 36)^=0, whence x = A sin ~2t + B cos 2 + C sin 6 + D cos 6*. Differentiating the first equation and subtracting three times the second to eliminate differential coefficients of y, we have dt *" whence we obtain the value of y without any new constants, viz. : y=-%B sin 2t + 2 A cos 2t + i&D sin 6 - ^- EXAMPLES. Solve the equations i. 2^-(i-*)y=**. 2. 'cte 3. 4. 5. (1- . 2 2 cosy 8. Obtain the integrals of the following differential equa- tions : + 9y - 25 cos ^ [I. C. S, 1804.] EXAMPLES. 277 9. Integrate the simultaneous system _ 4=0. 10. Find the form of the curve in which the tangent of the inclination of the current tangent to the #-axis is proportional to the product of the coordinates of the point. 11. Find the form of the curve for which the curvature varies as the cube of the cosine of the inclination of the tangent to the 12. Show that in the curve for which the projection of the radius of curvature on the 7/-axis is of constant length (l) S oclogtan(?+|), (2) y oc log sec ~. ANSWEES. CHAPTER I. PAGE 12. 1. Area = e 6 -e a . 3. Area=ia 2 tan 0, 4. Vol.^. 5 2. Vol. =-(e* b -e 2n ). Vol. = -a 3 tan 2 <9. 5. Vol.=f7ra 3 . 6. (a) Vol. =| Vol. = 1 TT 1 * VoL = (8) - JL 25 u Vol. =JL t) 7, 7Tfia 3 . 8. Mass of half the spheroid = J?r/xa 2 6 2 . ANSWERS. 279 CHAPTER II. PAGE 17. 2. ^Y. 6. 1. 10. a + (sin 6 -sin a). 3. ?^1. 7. x/2-1. 4. Iog e |. 8. |. PAGE 23. 2 r !00 r !000 r !001 X ft X Ht; & ' *' C> loo' Tooo' Tool' _ c .o _^ _^ 10' 100' 98 PAGE 25. , 2 J 2. a log x, ~j a log ^ +#, PAGE 26. 2. 3. logtan" 1 ^, logsin' 1 ^, log(log^). 280 INTEGRAL CALCULUS. PAGE 28. 9 4- aT + I " log2 J 4 + log 3' log 6 + logo 2 ' 4. log tan ^, log sin ^ - cosec ^. an-^, l 86 ^! >> _ ? 5. sin- 1 *, 1 tan-, 7. 8. -log^ + e*), log(log sin x\ CHAPTER III. PAGE 32. 1. sine*, sin# n , sin(log^). 3. asin^+-tan- 1 ^ 4 , -a cose* +6 log cosh #. 4 4. J_ tan- x -JL-. 6. ~. 8. sin-V^- V2 2 ^2 3 _-, . L V* ' v PAGE 41. /- - g - - 1 ?, - -H-.-f Jain- 1 *, 2. cosh^+l), si ANSWERS. 281 3. -Vl=F, x/^, s 4. i(^ 2 +1)4, 6. I siii- 1 ^-2\/r^-iWl -^ 2 , | sinh- 1 ^ + 2\/l + # 2 + 7. xyJ\^3?, 4 cosh- 1 - + ^?x/5?^ 2 2 8. ^logtan^, -logtan a3 + 1 20#)cosh x - 5(^ 4 + 1 2^7 2 + 24)sinh x. 6)sin x, _ 84\V 2/ \ 2 4 J{2(2^ 3 - 3^)sin 2^ - (2^ 4 - 6^ 2 + 3)cos - 5?r 4 + 607T 2 - 240, 265e-720. PAGE 52. 1. (a) (m 2 + l)~^ rne cos(^-cot- 1 m), where #=sin#. (6) gsmg + cos^- sn -g, where A - = S i 4 \ 3 3 / / \ / \ ^7 4 1 , _i .tT 3 3^7 (c) ^ tan ^ + log cos x. (e) - tan l x - (d) ^tan" 1 ^ Jlog(l+^ 2 ). (/) x sec" 1 ^ - cosli" 1 ^. (a) x \/l ^sin" 1 ^. (b) 6>(sec(9 + cos0)-sin<9-logtan + Y where ^ = sin(9. (c) 2 (c?) (sin <^> cos ), where ^= ANSWERS. 283 3. (a)!*"-*. W *p^. m l+m 2 v 7 l+^ 2 (6) ie*/I + - A^cos^tf-tan- 1 \\ where tan0 = #. ' 2 \m Vm 2 + 4 V m//' (c) -e me ( JL_c 4 Vm 2 +l where tan B=x. 4. (a) e*sin^. (6) ie*(#- 1)- x v (a sin 6.^7 sinh ax b cos u? cosh a^) 2 2 (d) -- 6~ ax \ sin(6^ + 2) V, where r and are as in Art. 53. r* ) (e) 2 X (P sin 2# - cos 2^), where = cos < - cos 2<> + cos 7* ?'- 7* =. sin > - sn r r 2 and r 2 = 4 + (log2) 2 and "log 2* +\/l^ 2 log + log _ ^ 6. x * x g 7. - cot (9 log(cos (9 + \/c7^2g) - 6> - cot (9 + AVC .. Sill C7 8. sin 6 cos log(l + tan 0) - - + ilogsin( + - Y 9. (a) e*tan-. (6) -e*cot-. 284 INTEGRAL CALCULUS. CHAPTEK V. PAGE 58. 2. . 3. \ log(^' 2 + 4# + 5) - taii" 1 ^ + 2). 4. -log(3-.r). 5. #-2log(# 2 + 2.r + 2) + 3tan- 1 (. 6. 2# - f log(# 2 + 6^7 + 10) + 1 1 tan- 1 ^ H- 3). PAGE 62. (iii.) ^ - _ ct o (ix.) x + 7 a ^T giZ^ log(^ - q.) +etc. - - j e (^Tiy + 8 (^-l) 2 8(^-l) + 16 B 1 l a^^ + I_L.-?-JL_. 27 g ^-l + ANSWERS. 285 3 - (ii.) . + ( 2 (iii.) tan- 1 *- 4 (i '> - * tan- 1 * VS. (iv.) ^ tan-'(4r -_) 3^2 \A/21 x*J (viii.) (v,) l 2V2 5. (i.) l 3 (ii.) log^/V^T). (iv.) (v.) -lo g (* (iii.) l 286 INTEGRAL CALCULUS. (ix.) ilog^ (x.) ^ o 6. (i.) ll 17 17 - 7 17 (iii.) , + l I ANSWERS. 287 CHAPTER VI. PAGE 68. 1. sinh-^+1, x/2 2. 3. bx + a) - ( 4. -LRcar - b)Jc(a + 26^ - cr 8 ) + (6 2 + acj PAGE 74. 1 / 2 \ 5 3 1 /sin 6^7 3 sin kx . 15 sin 2.r n n 288 INTEGRAL CALCULUS. 2H 7 5 O 1 or cos x + co A - - cos 5 ^ + - cos 7 #. ( - l) n /sin Znx _ 2^/7 sin(2tt - %)x , 2M @ sin(2?a - 4)^ _ ) 1 2^-2 2 2^-4 "*/' COS(2tt-l)#_ 2M+1 ff COS(2^- 3>, \ l 2n-l 2n-3 "f or - 2. Jsin 3 ^-^sin 5 ^, - sin in 2# - 2 sin x- sin 6^ + J sin 3. Jtan 3 ^, 4 7r ~ 2 43 157T + 44 8 ' 60^/2' 192 5. \ cos 4 #, f sin 2 ^ | sin% + f sin e #, - cos nx - r^ . cos(n _ 1. 2\/tan#. - - 4(n-2) PAGE 83. 3. (i.) [a0 + 6 log(a cos (9 + 6 sin (9)]/(a 2 + b 2 ). (ii.) [(ac + be)0 + (be - ae)log(c sinO+e cos (9)]/(c 2 + 6 5. (i.) 1 tan- 1 /" a tan A 0*1 a* -V* Wo^-P / / \ 2, i/l, ^\ /- \ 1 -u i (11.) ^tan- 1 -tan- . (m.) -: cosh- 1 ' ' 3 \3 2/ 7 sma ANSWERS. 289 (iv.) ^-i3co^-tan-i3)-x/10 S-vlO 000(4?- tan- z 3) ( vii.) a; cos a + sin a cosh- 1 1 + cos a CO8 x . cos a + cos x (ii.) ^v^CTW&^-V^tan-'-tan- 1 ^^) }. (iii.) _1_ .( -' if a>a' and a'b>ab', with analogous forms for other cases 8 - ' - 3 3 (l+2cos(9) 2 ' Q sin 2^, ^.cos^ + sin^ 1, y. log - _ - . W sec 20. - r AW o 7i ?i ^ 2 cos # sin 2 10. cosh x tan-. 11. - 2i 12. - 2 x/1 - sin x - \/2 log tan(|+| V 13. 4 1 - sin ^ 2 1 + sin # v 6 a 15. -. 18 . K I. C. T 290 INTEGRAL CALCULUS. 2 16. log log tan x. 19. na 2 17. -cosh-^cosfl+sinfl). 20. cos x + x sin x 22. 23. 24. cosec- 1 (l+sin2^). 25. 26. artan-^-logtl+tf 2 ). 5 - tan x 1 8 CHAPTEE VII. PAGE 94. 7. If ANSWERS. 291 and /i= _ r xax x^f . 5 / 2 = _ -- +-a T x\2ax-xrf , 7 7 3 = -g- -+-a Between limits and 2a, A = ~^ / 2 = f 4 , /3 = PAGE 95. 1. fsi and similarly for 2, 3, 4, 5. 6. fsirAEcfe= _co^ 8 in% + 3/ ; _ g in2 f \ an / 4 4 \ Zi 4 / / sin 6 ^ dx, etc. t, T 7 sin^cos"" 1 ^ nlf <> 7 7. / cos 71 .! 1 c?j; = - + - I cos w ~ 2 ^ ax. J n n J 8 / \ _ sin 3 ^ cos% 1 f _ si 6 2\ sin x cos 3 # 1 / , sin 2^\ \ _ , 4\2 4 // (ii.) su - x - - 3(x sin x cos #). (iii.) tan x 2 cot x \ cot 3 #. cos x PAGE 102. TT 3?r 35?r 128 g 3?r 8 8 1 4' 16' 25(3' 315' " 2 9 ' 693' 693' 60' 4. J siii 8 ^, i sin 8 6> - r a o sin 10 6>, J sin 8 6> - J cos 3 ^ + f cos 6 6> - f cos 7 (9 + J cos 9 0, - JQ sin 5 6> cos^ + yjgd ^ - i sin 40 + ^ sin 8(9). 5 ^28-71^/2 37T-8 3?r + 4 289 ?r 2 5?r 1680 ' 32 ' 192 ' 4480' ^ 8' 9' 192* 292 INTEGRAL CALCULUS. PAGE 104. 2. If I m< n denote the given integral, m + n+l 6. With a similar notation, / \ T (K n (a^bxf^ an T W n>p ~~ n ,v,m- (y) (m - n + l)/ m , = - - (m - 2)a 3 / m _ 3 , " 1 (3) m/ m =^- 2 (^ / = __^ cos 3 Xa cos x + 4 sin #) 2 + -g-^ 2 1 cos X acos ^+2sin^)+2. 1 . - 8. I n = - ,- 7 _ . i a sin a? n cos # 7 n = ^ 71 - 1 . i a sin ^7 sin eu? + w cos ^ cos ax n(n 1) r I H = -8lll n - l X - 5 - - 9 - -- 1 -- ^ - win-* n 2 - a 2 n* - a 2 17 1 , ___ ra , w(m~l) ' 3m 3m(3m - S) 3?7i(3m - 2)(3m - 4) "*" -2).. .(m + 2) ANSWERS. 293 ork , x ram- 20 (77? PVPTIl v ; a a >rn 9VH Aj2_t_92\ III ) /...^/t -TAJ It 2 oosli ' m(m- 1X^-2X^-3).. .3. 2 23. /^ 2 sm(tt-l)* 7 n - 1 28. (n-l)u m = _ 29. aml m + (2m Q1 / N r r 31. (a) / n = / n _ 2 - (/?) Deduce from 25. / v r _ _(7i- 2)^7 cos ^7 + sin ^7 , w 2, (^_ l)( 7i _2)sin"- 1 ^ + ^T n CHAPTEE VIII. PAGE 115. \/3 2. -cosech- 1 ^, - - x/2 v/^ 2 +2^+3 - sinh- 1 - -- 3. 1 lo 2\/2 ^- ^^+l JL \/2 294 INTEGRAL CALCULUS. _ x/2 f _ 7=tan PAGE 117. * 2 ^=^g A 2 --^ sinh -ws 3. V2 5. If a, 6, c are in ascending order of magnitude with modifications for other cases. 6 p 1" _ 9 sinh- T9 PAGE 120. COS 07 - COS 1. - -log cos ^ + cos ~ 6 2. 2(sin#+#cosa). 3. sir 4. Prove (TZ. being a positive integer) cos nx cos wo, r, r ~-\^ / \ , sin Tia ^u^-u, _ 2 cogec a ^ gin ra cos (^ _ r \ x + __ . cos x cos a r=i sin a / Then cos a? cos a sin a r= i ?i ANSWERS. 295 5. 2 sin x + 2 sin a log(sin x - sin a) -f 2 sin a log tan -cot tan --tan" 6. 5 _ 2 2sm 2 a V 2 PAGE 129. 1. (i.)2tan-V#. (iii.) -4- \/2 (ii.) 2tan- 1 v/fT2^. (iv.) -sinh- 1 -- z. x/3 l (v.) v/^+^+1 -l sinh- 1 5^+1- sinh-i-L Llf \/3 (vi.) ^^L 1 . (viii.) (vii.) -- L B inh-if2 no? \*' (ii.) If a>c cosh-i e. with a corresponding real form if a < c. 3. (i.) v (cos a cos /5)(cos a - cos y) 2 1 X mall -1 CQS ^ + COS a COS a ~ COS ft cos a ~ C QS 7 "~ cos a cos cos a cos y for the case cos a > cos fi or cos y with modifications for other cases. 296 INTEGRAL CALCULUS. _1 Vsin(a /3)sin(a - y) 2 -+, l n tan 3? cot a cot a cot 3 cot a cot-/ x cosh" 1 -: : t . cot /? cot a cot y - cot a 6. (i.) j iogX24 (ii.) -L. (iii.) Mi) 5^5. (ii.)ilog2. (iii- 14. (i.) Iog e 2-l. (ii.) . 15. (i.) JL. (ii.) 1. 27. (i.) Iog e 2. (ii.) F. (iii.) | 29. 2/^. 30. e- 1 . CHAPTER IX, PAGE 141. 7. (i.)^-^). (iii.) 5 (ii.) (rj-r^^S (iv.) 2a(cos|-cos|^. 12. *L. PAGE 151. a * a-x 2. 8a. 3. The Cycloid. ANS WERS. 297 r\/l+3cos 2 0_ V3, v/1 + 3 cos 2 + V3 cos ff~\ e * ~~co^0~ --2- 10 ^^^ - 5. 5o. CHAPTEB X. PAGE 158. 2. (a) c 2 sinh-. G (b) e h -l. M (o) ^ - g V^rp _ * sin-i^i 2 (/) Mog 5. 4a 2 . 6. 37ra 2 . PAGE 160. 2. * 3. 2 . 8 16 4. ^. Total area=^! (w even), or * (n odd). 4?^ 2 4 5 2 tan a 2 /3 cot a/^gycot a _ -i \ R a 2 /3 3 -a 3 4 ' 6" ~^g3- ' 8. z. 9. . 2 a/5 2 PAGE 167. 1. 37ra 2 . 3. -(tan f + J- tan 3 i/r). 2 298 INTEGRAL CALCULUS. 4. PAGE 178. 1. T V* 2 - 2. ^. 5. (7r-2)a 2 . 7. 9. (i.) %nrab. (ii.) 128 r(m + l) m being supposed odd. 11. a 2 [2log(\/2 + l)-iW2l 14. -(32 + 24\/3 - STT). 6 12. 4cV2sin- 1 -L- 17. ^?V2. v/3 22. - 23. ^7? - ! )- 24 ^ even ' nr ; 7 ' odd ' ^ 2 ' 25. 7r^-^". 30. ^^-. 39. ^-. 12 4 4 28. TT( 6 2 + J. 32. a 2 ( 1--J. 40. 7raj(a-b). 29. ^(107r + 9 x /3). 34. 2 (7r + 2). 41. a 2 . 42. 7rV 2 - i*s4 bm 44. (i.) , (ii.) i ANSWERS. 299 CHAPTEE XI. PAGE 187. 3. 7 PAGE 191. 2. 3. If the sides be a, 6, c ; s the semiperimeter ; and h^ h^ h 3 the distances of the midpoints from the given line, surface = 27r(aA 1 + bh 2 + cA 3 ), volume = -^(hi + h 2 + k 2 )>Js(s - a)(s - b)(s - c). 3 PAGE 193. 1. If a = rad. of base, 5. 27ra 3 (log e 2 - ). h = altitude, 6. Surf ace =% 2 -7ra 2 , = slant height, volume =irV. surf ace = Tral, 8. J7r 2 a 3 . volume = ^TraPk. 10. T V7T 2 a 3 . CHAPTER XII. PAGE 201. 1. (i.) 'ass=-g-, (11.) x = y=^ ( m -) 2 f i ) M- - (ii.) J=l a) y = 2 Z . P Z. (iii.) 300 INTEGRAL CALCULUS. 3. f of length of rod from end of zero density. If ^length, . 5' PAGE 207. 1. Let 20 be the angle, a the radius, the median the initial line, 2. 5 = 2, about tang., n+4' about diam., 4. If^ 1= _A^3, and m m 2 w 3 m Mom. In. about ^-axis = 6. Area = _. 2(3v/3-7r)' Mom. In. a 3v'3-7T 7. (1) f 9 ^, ??V (2) Hijfa 2 , (3) \ K R / ^^ a (i) jr^, (2) (2) (a) ANSWERS. 301 CHAPTER XIII. PAGE 215. 1. x tan x - log sec x =y tan y - log sec y + C. - 3 3. 5. 6. 9. (1) y = (7e (3) (2) ?y2 = 2^ + a (4) 10. * PAGE 219. 1. 2ye iAn ~ lx ==e 2i&n ~ lx +O. 2. (a 2 + b' 2 )y = a sin bx b cos bx + Ce~ ax . 3. rO = a + C. 1 -I o. X6 ^= tan y -f~ C/. 1^2. - ; "{"C/. x sin ?/ 2^7 2 6. ye 2 v*= 2V^7 + a 13. - = ^L+ (7. x log 2 2^? 2 2 9. -L_JLf(7. 15. i = i + Cfe' ^cy 2.^^ 7* a (7. 16. JU= -- 302 INTEGRAL CALCULUS. 18. (1) f^ 1 +C-. x 2# 2 (2) (a 2 + & 2 y = a sin bx-b cos 6^ + (7e (3) si CHAPTEE XIV. PAGE 223. 2. - = 2x/73 where v=yjx. 3. i-I = a ^ y 4, The jo-eliminant of y= and + 0} _ 7- (B- I) 2 \/4^6 Y -(^- I) ' PAGE 226. 2, (y-^) 3 4. 5. .r- 6. Gy-S^-^ 7. 8. ANSWERS. 303 PAGE 230. 3. + C, 4. o~s 5. 4 6, c< PAGE 232. 2. y = Cx + C\ 3. y = 4. y = logp-p + C - PAGE 233. 5. ?/ = 2.y=apx+p\ ] p n x = (n- 6. v = 1 ^ JL 3^ 3a ~ 2 p I 3a 2 8. A rectangular hyperbola. 304 INTEGRAL CALCULUS. 9. Parabolae touching the axes. 10. Hyperbolae. 11. A four-cusped hypocycloid x*+y*' =0?. 12. 8?/=(2^-l) 2 . 13. y = y=c cos 3 6> , 3c sin 1 14. 2/ 2 =& 2 - -jTpj a series of conies touching the four straight lines x*J^Ay = \A#, the singular solution. CHAPTEE XV. PAGE 238. C dy 1. y=x\ogx+Ax + B. 6. x+b= \- - = J 2. y = acosh(- + 6\ 7. 3. 2y = 4. ,-^ +& . 9. y = 6 tan /1 5. (^-^)2 + ( y -J5) 2 =a 2 . 10. 11. y = Ex* 1 Ax log x. PAGE 242. 2. 3. (a) (o) ^3-^2+? = ANSWERS. 305 CHAPTER XVI. PAGE 251. In the results of the following set all capitals denote arbitrary constants : 1. y = Ae ax +Be bx . 3. y= Ae x + fie** + Ce 5x . 2. y=*Af*+ Be** +&"*. 4. y = ( cos 2 8. y = A sin a? + B cos a? + <7e~ f sin *? + Z)e~ f cos 9. y = (A + 0)siii ^P + ((7+ 7>o?)cos 10. y = (-4 + ^ + Cfeajsin a? + (Z) + JEr + ^r 2 )cos a? 11. y(44-^+^^+l)^4g r2 sin(log ^7) 2 cos(log a?) log x cos(^ log ^ ~~ 4. - 5. y = ^! sin -j | log(a + bx) \ + J 2 COS | T lg( a + ^) [ 308 INTEGRAL CALCULUS. CHAPTER XVII. PAGE 269. 3. r = be~ e ^ a . 4. = l-cos<9. PAGE 276. 2 2. Put tany=2 ; tan?/ = t a + bx=S; y = C( where m^ m 2 are the roots of the equation 2 * * 3. Put a + bx=S; y = C(a + bx)^ + D(a + x* - + .., no b{n-\-Ao) 4. Put 2 = tan~^ ; y = 5. Put = sin~ 1 o? ; y=^ 6. Put 6^ = ^, ^=77; (<*-e x +iy*=A. 7. Put sin^=^, siny=?7 ; (siny 8. (a) y = (b) y = (c) y ^^ 3 sin(log x) + JB^cosQ.og x). 9. y + Z = A sin 3x + B cos 3x + C si 32= - 6(^1 sin 3x + B cos 3a?) + (C7siii 4r + D cos 4^). 10. = Ad\ 11. 2/ = ^ 2 GLASGOW I PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. WHICH BORROWED ^ Publication 'Fdue on the LAST DATB and HOUR stamped below JUL 7 j5 Cs .,. (F5759slO)4188 T n . e 'al Library niversity of California Berkeley ASTRGitiGf".;