^ Irvi 
 
 ng Stringhan 
 
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^ , 
 
M 
 
^- 
 
 SHORT TABLE OF INTEGRALS 
 
 
 
 COMPILED BY 
 
 B. 0. PEIRCE 
 
 HoLLis Professor of Mathematics and Natural Philosophy 
 LN Harvard University 
 
 BOSTON, U.S.A. 
 GINN & COMPANY, PUBLISHERS 
 
 1903. 
 

 The compiler will he grateful to any person ivho may send 
 notice of errors in these formulas to 
 
 B. O. PEIRCE, 
 
 Harvard College^ Cambridge. 
 
 IN MEMORIAM 
 
 •i U 
 
I. FUNDAMENTAL FORMS. 
 
 ax. 
 
 1. ladx = 
 
 2. I af{x) dx=:a if(x) dx. 
 Q rdx I 
 
 +. I a;"*da; = , when m is different from — 1. 
 
 J m -h 1 
 
 5. \edx = e^. 
 
 10. , _|^- .= yersin -'*. f ^ ■■ =• - cS«•^i^X ,. 
 
 Ai» I COS a; aa; = sin a;. ^ 
 
 / 
 
 I2» I Uin x dx = — cos x. 
 
 86B425 
 
^ '; ..; ' 'FtridAMENTAL FORMS. 
 
 15. ■rtana!seca;(to = seca;. 
 
 16. j sec'' a; da; = tan a;. y^ 
 
 17. )csc'a;da! = -rtna;. J 
 
 I„ the following fo.™nlas. «, ., «-, -d y represent an, 
 functions of X'. 
 
 19a. Cudv=^uv-ydu, 
 
BATIONAIi ALGEBRAIC FUNCTIONS. 
 
 II. RATIONAL ALGEBRAIC FUNCTIONS. 
 
 A. — Expressions Involving {a-\-bx). 
 
 The substitution of y or z for x, where yz=xz = a-^ fee, 
 gives 
 
 21. j\a-j-bx)'^dx = ^ Cy^'dy. 
 
 22. (x{a-}- bx^dx = -^ fy^'iy — a)dy. 
 
 23. (iir(a + bx)'^dx = -^^ i y"" {y — tiYdy. 
 
 J x'^ia + bx)""" a'"+" V 2'" 
 
 Wheuce 
 
 26. C-^=.\\og(a-^bx), 
 J a-\- bx b 
 
 27. r ^^ = ^ 
 
 J (a + fea;)2 6 (a + 6a;) 
 
 28 r ^^ =: ^ 
 
 ' J (a + &a;)'* 26(a + 6a;)2 
 
 29. r_£^ == 1 [a + 6a; - a log(a + bx)\ 
 J a-\- bx b^ 
 
 J {a-hbxy b'l ^^ ^^a + bx] 
 
D RATIONAL ALGEBRAIC FUNCTIONS. 
 
 ' J (a -h bxy b' [_ a + &a; 2(a + bxyj 
 
 32. r_^^ = 1 [^(^ + 5a;)2 - 2a(a + bx) + cr log (a + 6a;)], 
 J a -\-bx 0" 
 
 S4. r_i^_=_iiog2^t*i^. 
 
 J ic(a + 6a;) a a; 
 
 35 
 
 J a; (a + 50?)- a(a-f-5a;) a^ a; 
 
 f__^^___ = _ J_ 4_ A log ^L±_^. 
 J x^(a H- 6a?) ax a^ x 
 
 36. 
 
 rfa; 1 ^ la; 
 tan ^ — 
 c c 
 
 ^^ =_Llog^ + ^- 
 
 B. — Expressions Involving (a H- 6a!f*) , 
 
 38. I - 
 
 ' J c^ — a;^ 2c ^c — a; 
 
 ^^- f— 7T3 = -^ta'^^'«^\l^' if a>0, 6>0. 
 
 *^- I , , 2 = .) / log-— -i: -^, if a > 0, 6 < 0. 
 
 41, C—^^^—— — ^' L JL r rfa 
 
 * J (a + 6a^^)^'~2a(a + 6a;2)"T-2aJ a + 6a;2* 
 
 4i> r <^a; _ 1 X 2 m — 1 / * (?a? 
 
 "' J (a + 6a:^)"*+i 27)ia (a + 6a;2)«"*" 2ma J(a + 6a2)' 
 
 Ja + 6x-2 26 \ hj 
 
RATIONAL ALGEBRAIC FUNCTIONS. 
 
 .. C xdx I C dz , - 
 
 '*• I 7 , o, ^, = - I r-: — -,, where z = or. 
 
 45, I = — log . 
 
 J x{a + ba^) '2 a a + ba^ 
 
 r oi?dx _ ^ _ « r dx 
 ' J a-\-bx- b hJa-\-ba? 
 
 »/ it'^(a + 6.«-') ax aJ a-^-bx^ 
 
 48. r '^c^^' _ —^ , I r dx 
 
 J (a + 60^)"*+^ 2?rt6(a + 6a;2^"' 2m6J {a-^-ba?)"" 
 
 J ay'ia + b3f)'^+' a J x^{a + bx^ a J (a + 6a;^)"'+i* 
 
 where bk^ = a, 
 
 51. r_^=XJiiog/^^lllMil^ + V3tan-^^^^'l 
 J a-[-ba? ^bk\J \ {k-{-xY J^ h^l ] 
 
 where bk^=a. 
 
 52. f:._^ = -Llog-^. 
 
 r da; ^1 r dx b r ^ 
 
 ' J {a-^bx'^y-^^ aJ(a-\-bx''y' aJ{a-\- 
 
 x^'dx 
 bx") 
 
 'dx 
 bx''y+^' 
 
 • J (^a-{-bx^y+' bJ {a + bx'^y bJ (a + 6: 
 
 r- r <^a; ^i rda; b T dx 
 
 J ar^a + bx'^y^^ aJ x'^(a + 6a;'')p aJ xT ""(a + 6a;") 
 
 IH-I 
 
RATIONAL ALGEBRAIC FUNCTIONS. 
 
 II 
 
 + 
 
 I 
 
 'h 
 
 I 
 
 II 
 
 a 
 
 03 
 
 II 
 
 o 
 A 
 
 a 
 
 > 
 
 ^ > 
 
 ^1^. 
 
 o 
 V 
 
 ?> 
 
 c 
 <v 
 
 1 1 
 > 
 
 + 
 
 I ^ 
 
 1 1 
 
 > 
 
 I 
 
 + 
 
 I! § 
 
 I ^ 
 I I 
 > 
 
 ^IX 
 
 ^|><l 
 
 'tS 
 
 '^ 
 
 II 
 
BATIONAL ALGEBRAIC FUNCTIONS. 1 
 
 64 r^dx __ _ 2a4-6g _ b(2n — 1) T^^ 
 
 65. fA'do^^^-JLlogX + ^ll^l^^r^. 
 J X c 2c' ^ 2c^ J X 
 
 "^ J X^ cgX qJx 
 
 ' J A""+^ (2n-m + l)cX" 2n-m4-l'cJ X"+^ 
 
 'dx 
 
 ■ m — 1 ^ a r^ 
 2n — m-f-1 cJ X 
 
 n+l 
 
 •Ja;X 2a ^X 2aJ X 
 
 rdx ^ b , X L4./'_^_^^^^ 
 
 'Ja;-X 2a2 ^a^ aa; \2a2 aJJX 
 
 70 r__^^_ = \ ??. 4-^1 — 1 6 r dx 
 
 • J aj'^X^+i (m - l)ait"»-iX»' m - 1 * a J x'^'^X' 
 
 2n-\-m — 1 c^ r dx 
 aJ a 
 
 D. — Rational Fractions. 
 
 Every proper fraction can be represented by the general 
 form : 
 
 /W ^ 9i ^""'-^ 92 ^''~' + 93 X''-' + - + f7„ 
 F{x) x^ + k^x"" ^ -{- ksx""-'^ -i \-k,, 
 
 a, 6, c, etc., are the roots of the equation F{x)=0, so 
 that 
 
 F(x) = (X- - ay (a; - 6)« (a; - cr •-, 
 
10 RATIONAL ALGEBRAIC FUNCTIONS. 
 
 then /W_ = ^4^ + ^^^— + A_+...+-A_ 
 
 F {x) {x - ay {X- ay ' {x - ay'' x-a 
 
 f— ^i— + ^ + ^ 4--+-^ 
 
 (x- by {x- by-' (x - by-' x-b 
 
 I ^1 _| Yl J ^3 _L. ... _J_ ^r _ 
 
 {x — cy (x — cy-^ {x — cy^ x—c 
 
 Where the numerators of the separate fractious may be 
 determined by the equations 
 
 ,^ (.) =«g^^ <^. (-) =^^^J^^ etc., etc. 
 If a, 6, c, etc., are single roots, then p = g = ?•=••• = 1, 
 
 ""•^ /(a--) _ ^ , J? , c 
 P" (a?) a; — aa; — 6a; — c 
 
 where vl — -^ ^^^ B — ^^^' eto 
 
 where -^ - p, ^^y ^ - p, ^py ^^- 
 
 The simpler fractions, into which tjhe original fraction is 
 thus divided, may be integrated by means of the following 
 formulas : 
 
 r hdx _ rhd(mx-\-n) 7i 
 
 J {mx-\-ny J m {mx -\- ny m {1 — I) {mx -}- ny~^ 
 
 C hdx h I X , >, 
 72. I == — log (mx + n) . 
 
 J mx-^n m 
 
 If any of the roots of the equation f{x) =0 are imaginary, 
 the parts, of the integral which arise from conjugate roots 
 can be combined together and the integral brought into a 
 real form. The following formula, in which i=:V— 1, is 
 often useful in combining logarithms of conjugate complex 
 quantities : 
 
 78. log {X ± yi) = i log (x2 + /) ± i tan ' •^. 
 
IKBATIONAL ALGEBRAIC FUNCTIONS. 11 
 
 III. IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 A. — Expressions Involving Va -|- hx. 
 
 The substitution ,of <a new variable of integration, 
 y = Va + bx^ gives 
 
 74. ( Va ■+■ hx dx = ^ ^/{a + bxy. 
 
 75. r«V^r+ted.- = -^i?iLz^5MVIH+M!. 
 
 J 1056^ 
 
 J X ^ ^yja-j-bx 
 
 r dx _ 2 V ft 4- 6a? ^ 
 
 C ^d^ '2(2a — bx) / . , , 
 
 79. I . = '^-TTi ^VA + 6a;. 
 
 80. f >-^^^ ^2(8a^-4a6.. + 36^a^)^^^-^:^ 
 
 81. r_^g^ = -JLlog/: -^'- f \ for a>0. 
 •^^ a' Va + 6a; Va VVa.+ 6a; + Va/ 
 
 ^^ r da; 2 . _ i la + ^^' i?^ ^ ^ rk \ 
 
 S2. I =: = tan ^^ — ' , for a < 0. ^ 
 
 Ja;Vrt + 6a; V — a ^ — « 
 
 r (Jx _ V a + 6a; 6 r dx 
 •^a^Vo^^" <*^ • -«^ -^Va + da; 
 
32 
 
 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 84. J{a + bx)^'idx = ^Jy'^-dy = ?-^ 
 
 hx) 
 
 (2 ±71) 
 
 85. \ x(a-\-hx)~^dx = —\ ^ — ■ ^ ^^ — ' '- — . 
 
 J ^ ' W\_ 4.±7l 2 ±71 J 
 
 J ^ x'^dx _ 2x"' Va H - hx 2 7?ict /* x'^-'^dx 
 
 ^'a + hx~ (27/i+l)& (27?iH-l)6J VoTfe^ 
 
 / da; _ _ Va + 5a7 _ (2??. — 3) & r da? 
 
 a'" Va + 6a; 0^-1) «^'" ^ (27i - 2) a J a^'^-VoH^ 
 
 88. f " + ^;)^ ^^ = 6 J(a 4- hx)'^d. + a f ^^ "^f ^"^^ ^^^ 
 
 86 
 
 87 
 
 89./- 
 
 da; 
 
 = 1 C clx -k C- 
 
 dx 
 
 X (a + bx) 2 '^'" a; (a + 6a;) 2 ^""^ (a + 6ic) 2 
 
 B. —Expressions* Involving Va.-^ ± a^ and \la-—x^, 
 
 90. \^x^± a' dx = i \_x Va;' ± «^ ± a- log (a; + Va;''^±a^) ] . 
 
 91. I Va'— ar da; = -J-[a;Va'—a;^ 4- a- sin"^-]* 
 
 92 
 
 •J; 
 
 dx 
 
 ^x'±a? 
 dx 
 
 log( 
 
 
 — cir»~l 
 
 94 
 95 
 
 1 !« I 
 
 -COS ^-» — - -i- 
 
 Ja;Va2±a;2 a \ a; / jx/iVx- «t »- 
 
 I - — = — dx = Va- ± ar — a log ^ \ l=^r^ — ^ 
 
 Va^ — x^ 
 
 dx _ 
 
 = -eos -— r=: ~- iec :^ 
 
 a;Va;2_^2 a a; O- o^ 
 
 •These equations are all special cases of more general equations given in the next sectioa 
 
97 
 
 IRRATIONAL ALGEBRAIC FUNCTIONS. 13 
 
 J X X 
 
 98. f . ^^^ = ± -J'oF±^, ' 
 
 100. fx Va^ ± o? dx = 1 V(ar^±a2)S. 
 
 101. I ic Va"'' — af' dx- = — J V(ct- — a;^)^ 
 
 102. Cy/(x' ± crydx 
 
 = i\ x^ix' ± d'y± ^^\/¥±d' + ^\og{x +y/W±d')^ 
 
 103. C'\/{a' — x-ydx 
 
 if /7~^ — i:^Vs I 3a*.r / -i r, , oa^ . _ix"l 
 
 = ^ ajV(a- — ar)^ + - va-— :r4-- sin^ . 
 
 [_ 2 2 aj 
 
 104. r ''^ ±" 
 
 105, 
 
 da; _ a; 
 
 V (a- - ar'p ~ «' -Ja'-a^ 
 
 xdx — 1 
 
 f— ^ 
 
 ^ -slid' 
 
 106. f , , _ 
 
 107. r--:^^=.=— i=. 
 
 108. rxV(ar^±aO«da; = J V(^^^^ 
 
 109. CxyJid'-x'Ydx^^ -\\l{a- — xy. 
 
14 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 110. I x^^/af± a^dx 
 
 = -y/{x'±ay If - {x\/¥±^' ± a2 log (x + V^T^)), 
 4 8 
 
 111. Ix^y/a^ — x^dx 
 
 112. f ^^^ = ^' V^^±^ ^F ^'log (X -t- V^db^n. 
 
 113. r ^^^ ^_gv,,2i:^+gL%ia-i^. 
 
 2 2 a 
 
 118 
 
 114. I =± ; 
 
 115. f ^^ =-:Zg^. 
 
 J a- a; 
 
 J ay^ X a 
 
 r x^dx _ ^ -X 4_iog(a;-l-V^±^V 
 
 119. f ^^^ = ^ -sin-^^. 
 
 C. — Expressions Involving Va -\-hx + cx^. 
 
 Let X = a + 6a; + ca*^, g = 4 ac — 6", and h=z~ In order 
 
 to rationalize the function /(a;, yl a + hx -\- ca?) we may put 
 Va ■\-hx-\-cx^= yl ±c\l A+Bx ± a;^, according as c is positive 
 or negative, and then substitute for x a new variable 2, such 
 that 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 15 
 
 z = V^ -^Bx -h ar — x, if r- > 0. 
 
 
 where a and yS are the roots of the equation 
 
 ^+jBic-a^2^0, if c<0 and -^<0. 
 
 — c 
 
 By rationalization, or by the aid of reduction formulas, may 
 be obtained the values of the following integrals : 
 
 20. f-^ = J^iogfVX 4- oj Vc + -^\ if c> 0. 
 ^ -^X, Vc V 2Vcy 
 
 a-i r da; 1 . _,/— 2ca;~6\ ... ^^ 
 
 21. I-— ==-— =sin M . if c<0. 
 
 22. r ^«^ ^ 2(2ca; + &) , 
 
 23. f_J^^2(2ca. + 6)/l_^^A 
 
 ^ X''\fX 2>q\IX \X J 
 
 24. r dx ^ 2(2cx-\-b^/X 2k(n — l) r dx 
 
 J yr-^nt (27i-l)oX^ 2n-l J irn-iJY-' 
 
 X«VX (2n-l)gX" 27i-l J X^-'^X 
 
 25. C^Xdx = i^^^±^l^ + ± f— - 
 J 4c 2kJ ^x 
 
 26. fxVXda. = (2-dLaVX/^ AA + 3 T^. 
 
 J 12c V 4:k^8ky^lwJ sjx 
 
 28. rx-vxda;= (^^^+^>^^^ + ^^+^ r^^. 
 
 J 4(n + l)c 2(n + l)kJ VX 
 
 '-^ ^/x'" c 2cJ Vx' 
 
16 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 130. C ^^^^ = '^(b^-h'2a) 
 
 -^ X" VX (2n- l)cX" 2cJ X"Vx' 
 
 r QiTdx ^ {2b' — 4:ac)x-{-2ab 1 f 
 
 dx 
 
 r x-dx ^ (26^-4ac)a;+2a6 4ftc+(2yi-8) 6^ T c 
 
 '-'X'^VX (2w-l)c^X»-WX (2n-l)cg Jx- v .v 
 
 ^^v vx~l^~r2'c^ 8^"3^y ■^U'^~T^P vx' 
 
 1 36. faj VXdoj = ^^^ - - C-s/Xdx. 
 J ■ 3 c '2 c J 
 
 1B7. (xX VXdo^ = ^^^ - ^ fxVXcia;. 
 J DC 2cJ 
 
 '•^ VX {2n + l)c 2cJ Vx * 
 
 J \ 6cJ 4c 16c^ J 
 
 V VX 2(n + l)c 4(7i + f)cJ VX 
 
 a r X^'dx 
 
 2(n4-l)cJ VX * 
 
 141. Ca^y'Xdx=(x--l^^^-^-^'\^:^. 
 J \ 8c 48c2 ScJ DC 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 17 
 
 .„ r c^a; 1 . ^f bx-\-2a \ .J. ^ 
 
 .43. I — — = --=sin U — ^ , if a<0. 
 
 ^a;VX V-a \xyJb'^-AacJ 
 
 ^ x^JX bx 
 
 45. r_^i__= Vx ^ 1 r dx b r dx 
 
 / ' dx _ \/X b_ r dx 
 ^-</ IT ax 2aJ r. 
 
 a^Vx ^^ '^^^ x\/X 
 
 dx 
 
 dx 
 
 J X 2J yjx ^ X 
 
 'J xyJX (2n-l)VX -^ a?yX 2 J VX 
 
 '•^ ^' ^ 2-^a;VX ^^WX* 
 
 r_^cZa;_ ^ 1 r x^-^dx _ b Px"" ^dx _ a raf'^dx 
 '-'xWX c J x«-WX cJx«VX C'^ x»Vx* 
 
 ra; "'X"da; ^ a^'^-^X^^VX (2n + 2m- 1) 6 r a7'"-^Y"(Za; 
 '•^ VX (27i + m)c 2c(27i + m) J ^x 
 
 (m— l)rt r o;"* -X"da; 
 (2?i + 7?i)cJ VX 
 
 52 f ^^ _ ^ V^ 
 
 V arX»VX (m-l)aaf"-^X« 
 
 (2yi+2m-3)6 ^ da; ( 2?i+m--2)c r da; 
 
 2a(m — 1) ^ x'"-^X'"\/'X im — \)a J x"'~^X*'\fX 
 
18 
 
 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 153 fX^'dx ^ X^-'VX (2n-l)b rX^-'dx 
 'Jx""\/X (m — 1)0;"'-^ 2(m — 1) J aj'^-^VX 
 
 (2?i-l)c rX^'dx 
 
 1 J ^-2^X 
 
 m 
 
 54. rv2 
 
 D. — Miscellaneous Expressions. 
 
 2 
 
 a 
 
 versin"^— 
 
 ^2ax — x^ « 
 
 *^ V2ax — „ 
 
 56. f ^^^ = + J^. 
 
 57. f ^^_= = _ J^S. 
 
 J (a; _ 1)7.^2-1 \a;-l 
 
 68. J Ji±5c?a; == sin-la; - VT=^- 
 
 ™ X — X 
 
 + (a - 6) log ( Va; + a + Vo; + 6). 
 
 ^ ^{x-a)( 
 
 2 Sin \ . 
 
 \i8-a 
 
 (fi-x) ^^ 
 
 61. f ^ = _l,sin-\^i^±M, 
 
 62. r-v/a + 6.rda; = — ^/(a + &a?)*. 
 ^„ r dx 3 8/^ — , , .^ 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 19 
 
 164. r-,-gg^^- ^^^^-f^^) ^(^Tw. 
 
 165. I — = — sec V— • 
 
20 J TBANSCENDENTAL FUNCTIONS. 
 
 IV. TRANSCENDENTAL FUNCTIONS. 
 
 cos X. 
 
 07. i sina^'dx* 
 
 G9. I sin'^it' da; = — | cos x (sin^a; + 2) . 
 
 <<). I sm"a:f/.r = 1 I sin" ^xdx. 
 
 J n n J 
 
 71. I cos x dx = sin x. 
 
 72. i Qos-xdx=i^^mxQo^x-\-^x. 
 
 73. j cos^fl7dcc = |^siu.T?(cos-a;-|- 2). 
 
 74. I cos'*ii'c/a;= cos" ^:); sin.x'-j —- I cos" ^xdx. 
 
 75. I sin X cos x dx = ^ si n- ic . 
 
 70. I sin^ii" iioa-xdx = — g (^sin 4a; — x) . 
 
 7.Jsi 
 
 77. i sin X cos"* a; fZa; = — 
 
 COS'"+^T 
 
 m + 1 
 
 78. I sin"'a;cosa;cfa; 
 
 I sin'" a 
 
 S^' 
 
 79. I cos"' a; sin" a; da; 
 
 m+l 
 
 cos"*-^^a; sin""^'a; 
 
 m 4- ?i 
 
 _^m-^ (cos"" 2 a; sin" a; da, 
 m 4- »<^*^ 
 
 «QA r m • n 7 sin'*~^^• cos" 
 
 180. I cos"* a; sin" a; da; = 
 
 J m + n 
 
 '^— — - j cos'"a;sin'*"^a!;daj 
 
TRANSCENDENTAL FUNCTIONS. 21 
 
 181 r<^'OS'"a:dU-_ c os"' "^ 07 m — n + 2 /' cos'" 
 
 .7 sin"a; (?i — l)sin'*'^a/* 9i — 1 %/ sin" ^oj 
 
 jg2 rc os'^xdx _ cos" *"^ X , m — 1 r cos"'~^xdx 
 
 J sin" a; (??i — ?i)sin" '.r m — nJ sin" a; 
 
 cos"Y''~a;Vr---^'^ 
 
 */ sin*;* cc cos" it* 
 
 1 1 . m 4- n — 2 /^ dx 
 
 1 m + ^ ? — 2 r ( 
 
 "^aj.cos"^a; ?<, — 1 •/ sin'"a;. 
 
 71—1 sin"* ^aj.cos"^a; ?<, — 1 •/ sin'" ic. cos" ^x 
 
 1 1 7?i + 72 -5- 2 /^ dx 
 
 72 -r- 2 r 
 
 — 1 J sin*" 
 
 85 
 
 771 — 1 sin"*"^ X . cos" ^^• 7?i — 1 J sin'"^^it' . cos" a; 
 
 dx 
 
 J ^ dx _ 1 coso; m — 2 /^_ 
 
 sin'" a; m — 1 sin"''\T 7)i — iJ sin'" "a; 
 
 ' C ^^ — 1 sin a; n — 2 / * da; 
 
 *./ cos"a; 71—1 cos"^a; n — iJ cos"'~^x 
 
 87 . I tan xdx = — log cos x. 
 
 88. I tan^ajda; = tan x — x. 
 
 89. rtan"a;da; = ^^^H^^-^- ftan" 2a;c?a5. 
 J 71 - 1 J 
 
 90. j ctna;c?a; = logsina!. 
 
 91. I ctn^a;da;= — etna; — a;. 
 
 92. rctn"a;da; = - ^^^^" ^ -- fctn" ^ajda?. 
 
 93. j seca;da; = log tan f j + - J» 
 
 94. j sec^a;da; = tana;. 
 
22 TRANSCENDENTAL FUNCTIONa 
 
 195. Csec^xdx= f-^. 
 J J cos"* a? 
 
 196. j csca;da; = log tanja;. 
 
 197. i c&c^xdx = — ctnx, 
 
 198. I csc'*fl?da;= I — 
 
 'Ja + b cosa; ^a^^i^^ \_a-i-b cosccj J I 4 ««y >^ j c#» 
 '6 + a cosa.' + V6^ — a^ . sma;~| =^fSn -^ 
 
 200. f— ^-^ 
 
 1 , r 6 + g cos a.' + Vfe'^ — a^ . sma; ~[ 
 ' !_^2 [^ a + 6 cos a; • J 
 
 + c sin a? 
 
 — 1 . _ir 6^4-c^ + <^(^ cosa;H-c sina?) n 
 
 Va^ — W — c^ LV6^+ c^ (a + & cosa; + c sina;) J 
 
 1 
 
 • log; 
 
 V6' 4- c2 - a' 
 
 [ 5^ + c^ + g (& cosa? + c sina;) + V6^ -\- (^ — a^ (b sin a; — c cos a;) "] 
 V6^ + c^(aH-6cosa; + csina;) J 
 
 201. j a; sin a; da; = sin a; — a; cos aj. 
 
 202. I a^sina;cfa; = 2 a; sin a; — (a;^ — 2) cosa?. 
 
 203. j x^ sin a;da; = {Sx^ — 6) sin a; — (a;^ ~ 6 a;) cosa;. 
 
 204. j af" sin a;da; = — a;"* cos a; + m j a;"*"* cosajda;. 
 206. I X cos a; da; = cos x-j-x sin a?. 
 
 206. j ar^cosa;da;= 2a;cosa; + (a;^— 2) sina;. 
 
 207. j a;^ cos a; (Za; = (3 a;^— 6) cosa;-h(a;^ — Oa;) sinaj. 
 
TRANSCENDENTAL FUNCTIONS. 
 
 208. I x"" cos X clx = x"' s\n X — m I ic"*"^ sin x dx. 
 
 209. f^lBI clx = L_ . !lE^ ^ _i_ f^2^ ax, 
 
 J x"" 711 — 1 a;"* ^ m — 1 J a;'"-^ 
 
 210. CS2^dx= ^— . 2^^ L_ C^^^dx. 
 
 J of' m — 1 a?*" ^ m — iJ aj"*"^ 
 
 211. C^J^dx = x--^ + -^--^ + -^.,., 
 J X 3.3! 5.5! 7.7! 9.9! 
 
 -^- Cao^x-i ■, x^ , X* x^ , x^ 
 
 212. I dx = \oa^x ..» 
 
 J X ^ 2.2! 4.4! 6.6! 8.8! 
 
 aid. I sin ma; smna/*aa;= — ^^ ^ — >^ — — — ^—, 
 
 J 2{m^n) '2{m-\-n) 
 
 214. Ccosmxcosnxdx= ^'"<^ " ""> + "'"^^^ + ">'" ■ 
 J 2{m — n) 2{m + n) 
 
 215. I Bm'^xdx== a; sin ^ a; H- V 1 —x^. 
 216.^ I cos ^ a; da; = x cos^^a; — V 1 — x^, 
 
 217. j tan^a;da; = a;tan^^a; — ^log(l +a;^). 
 
 218. I ctn~^a;da; = a;ctn~^a;4- Jlog(l -f- a;'0* 
 
 219. I versin~^a;da;= (a;— 1) versin'^a; + -sj'lx — x^^ 
 
 220. j (sin~^a;)2da: = a;(sin"^a;)^ — 2a;4-2Vl — a;^sin"*aj. 
 
 221. j a;.sin"'^a:da; = :^[(2a.*^— 1) sin^^a; + a;Vl — a;"^]. 
 
 ««^ r » • -1 7 aj^+^sin-^a; 1 Cx^'^^dx 
 
 222. I aj'^sin ^xdx= I 
 
 J n-\-l ?i + 1 . ' V I - Jr 
 
 «^.» Cn -1 ^ a;"+^cos"^a; , 1 /".f"+*da; 
 
 223. I X" cos^ a; da; = 1 I — » 
 
 ./ w + l w + U Vl — a;^ 
 
 «^. C n, I ^ a;"+Han ^^' 1 fa^^+i* 
 
 224. Ia''*tan ^xdx — I 
 
 J n-\-\ n + lJ H- 
 
 dx 
 a?' 
 
24 TRANSCEKDENTAL FUNCTIONS. 
 
 225. j log X dx = X log X — .'>;. 
 
 220. ril2g^^c^.:._JL(iog.^)«.i. 
 
 227, I 
 
 t/ .T logo; 
 
 228. (- 
 J X 
 
 dx , 
 
 = log, logic. 
 
 dx 
 
 {\ogxY (7i-l)(loga^)"-i 
 
 229. i a;™ log ic f/a; = o;-^ i Tl^^ ^— 1 ( ^»* L ( * « ) - ^'*' k 
 
 230. re"-f?a; = — . i_ [^^ 
 
 231. Cxe'''dx==^(ax-1). 
 
 232. ra;-e-d^ = £l!£!-.!'^. fo^-^e-t^a^. 
 *^ a aJ 
 
 233. r^d^ = 1_^+_JL_ C_^a^ 
 
 J xr m-la.— i^m-lJx- 1 
 
 234. fe^^ Xo^xdx = ^!!i2S^ _ 1 f^!! ,a- 
 ^^ a aJ X 
 
 235. f V-' ^.-n o^^.. - ^"" (« sina^ - coso;^ 
 
 ^ a^ + l • 
 
 236. fe-cos:crfa; = £l(^LS£i^±_SH^. 
 
DEFINITE INTEGRALS. 25 
 
 DEFINITE INTEGRALS. 
 
 2.S7. f -^^_ = '', if a>0; 0, if a = 0; -'', if a < 0. 
 
 Jo cr + x- 2 2 
 
 238. j ic"-ie~''da?= j log- dx = T{n). 
 
 T{n+l) = n'T{7i). r(2) = r(l)=l. 
 
 r (?i -f- 1) = w !, if n is an integer T {^)— ^/tt, 
 
 Jo ^ Jo (l-fa;)'"+" r(m + w) 
 
 IT IT 
 
 240. j sin''a;da;= | cos'^xdx 
 Jo Jo 
 
 s=__L_: — "•K'^^— — ; . ![ jf jj ig ajj gygjj integer. 
 2.4.6. ..(w) 2 
 
 _ 2.4.6...(yi-l) .^ ^^ jg ^^ ^^^ integer. 
 1.3.5. 7. ..ii 
 
 — i /^ V / , for any value of n, 
 
 r-sinmxdx^^^ if m>0; 0,ifm = 0; - |, if m < 0. 
 
 Jo a; 2 - 
 
 r-smx.eosmxdx^^^ if m<-l or m > 1 ; 
 
 Jo X 
 
 ^, if m = -l or m=l; ^, if -l<m<l. 
 4 ^ 
 
 241, 
 242, 
 
 2^g r sirrxd x _'7r 
 
 \h x" ~2 
 
 244. r* cos (a^) da; = f* sin (a;^) fia; = i J|. 
 
26 DEFINITE INTEGRALS. 
 
 Jo l-{-x' 2 
 2-« ■ r'*' Gosxdx _ r°^ sinxdx _ B 
 
 Jo ^x ^0 V^ \2 
 
 247. ' ^^ 
 
 
 Vl — A;'-^ sin^a; 
 
 =l[i + (i)^^+6:!>'+ (lxlj^^+ ••■} « ^< 
 
 = /r. 
 
 248.J^ -s/l-k^sin^x.dx 
 
 ■.i[:-<«-*--(HJf-(l±5)-f-...}„^< 
 
 249. 
 
 Jo 2a^ 2a ^^ 
 
 Jo a"+^ a''+^ 
 
 Jo 2"+^ a" \a 
 
 252. r.-2c?.=^:!:^. 
 
 Jo 2 
 
 253. f e-«*cosma;da; = — -^ — -, if a>D. 
 Jo a^ -f- m^ 
 
 254. I e'''^8m'mxdx= ■ ;,, if a>0. 
 
 Jo a- + 7R- 
 
 ft2 
 
 255. I e-°'''^cos6a;da;= V^-^ ""' . 
 
 2a 
 
 256. fMf^d^ = -ZL' 
 Jo 1 - a? 6 
 
 267. fM^d^ = -^. 
 Jo l-{-a? 12 
 
DEFINITE INTEGRALS. 27 
 
 268. r^^dx = -^. 
 Jo 1—x^ 8 
 
 259. riog/^i±^V- = ^. 
 Jo \l-xj X 4 
 
 261. f '^^ =./i. 
 
 262. rVlogAY,^^=^(" + 0. 
 Jo =\^a;y (TO + !)»+' 
 
 263. I logsina!da!= | logcosac?a; = — - • log2. 
 
 ic . log sin oj da; = — — log 2. 
 
28 AUXILIARY FOKMULAS. 
 
 AUXILIARY FORMULAS. 
 
 The following formulas are sometimes useful in the reduction 
 of integrals : 
 
 266. logu = \ogcu -\- a constant. 
 
 266. log(— ^^) = logu -f a constant. 
 
 - sinWl —u^ + a constant. 
 
 267. sin^^w = ■{ —^sin~^{2u^— 1) +a constant. 
 ^sin~^2i* Vl —u^ + a constant. 
 
 > — tan^ + a constant. 
 
 268. tan-^?«=^ ^ 
 
 tan"^ — — h a constant. 
 
 L 1 — cu 
 
 269. log {x ± yi) = ^log {ay^ + ?/) ± itan"^--- 
 
 X 
 
 270. sin~^t^ = cos~Wl — u^ = tan^ = csc~^ — 
 
 Vl - u- '" 
 
 il 
 = sec^^ - 
 
 271. cos~^?* = sin Vl— ti^ = tan ^\ — 
 
 272. tan^^aj ± tan^2/ = tan V-^^ — ^V 
 
 Vl q: xyj 
 
 273. sin~*ic ± sin" ^2/ = sin^^ (xy/l —y^±yy/l —x-). 
 
 274. cos~^ X ± cos~^y = cos ~^ (xy ^: y/{l — x^){l —y^)). 
 
 275. sma;= --— 
 
 2i 
 
 e'* 4- e~*' 
 
 276. cosa;= '- 
 
 277. sina:/ =^i(e'' — e"*)= isinha;. 
 
 278. cos xi = ^ (e^ + g-^^) = cosh x, 
 
 279. log,;r = (2.3025851) logioa;. 
 
TABLES. 
 
 29 
 
 The Natural Logarithms of Numbers between 1.0 and 9.9. 
 
 N. 
 
 O 1 
 
 ! 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1. 
 
 0.000 
 
 0.095 
 
 0.182 
 
 0.262 
 
 0.336 
 
 0.405 
 
 0.470 
 
 0.531 
 
 0.588 
 
 0.642 
 
 2. 
 
 0.693 
 
 0.742 
 
 0.788 
 
 0.833 
 
 0.875 
 
 0.916 
 
 0.956 
 
 0.993 
 
 1.030 
 
 1.065 
 
 3. 
 
 1.099 
 
 1.131 
 
 1.163 
 
 1.194 
 
 1.224 
 
 1.253 
 
 1.281 
 
 1.308 
 
 1.335 
 
 1.361 
 
 4. 
 
 1.386 
 
 1.411 
 
 1.435 
 
 1.459 
 
 1.482 
 
 1.504 
 
 1.526 
 
 1.548 
 
 1.569 
 
 1.589 
 
 6. 
 
 1.609 
 
 1.629 
 
 1.649 
 
 1.668 
 
 1.686 
 
 1.705 
 
 1.723 
 
 1.740 
 
 1.758 
 
 1.775 
 
 6. 
 
 1.792 
 
 1.808 
 
 1.825 
 
 1.841 
 
 1.856 
 
 1.872 
 
 1.887 
 
 1.902 
 
 1.917 
 
 1.932 
 
 7. 
 
 1.946 
 
 1.960 
 
 1.974 
 
 1.988 
 
 2.001 
 
 2.015 
 
 2.028 
 
 2.041 
 
 2.054 
 
 2.067 
 
 8. 
 
 2.079 
 
 2.092 
 
 2.104 
 
 2.116 
 
 2.128 
 
 2.140 
 
 2.152 
 
 2.163 
 
 2.175 
 
 2.186 
 
 9. 
 
 2.197 
 
 2.208 
 
 2.219 
 
 2.230 
 
 2.241 
 
 2.251 
 
 2.262 
 
 2.272 
 
 2.282 
 
 2.293 
 
 The Natural Logarithms of Whole Numbers from 10 to 109. 
 
 N. 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2.303 
 
 2.398 
 
 2.485 
 
 2.565 
 
 2.639 
 
 2.708 
 
 2.773 
 
 2.833 
 
 2.890 
 
 2.944 
 
 2 
 
 2.996 
 
 3.045 
 
 3.091 
 
 3.135 
 
 3.178 
 
 3.219 
 
 3.258 
 
 3.296 
 
 3.332 
 
 3.367 
 
 3 
 
 3.401 
 
 3.434 
 
 3.466 
 
 3.497 
 
 3.526 
 
 3.555 
 
 3.584 
 
 3.611 
 
 3.638 
 
 3.664 
 
 4 
 
 3.689 
 
 3.714 
 
 3.738 
 
 3.761 
 
 3.784 
 
 3.807 
 
 3.829 
 
 3.850 
 
 3.871 
 
 3.892 
 
 5 
 
 3.912 
 
 3.932 
 
 3.951 
 
 3.970 
 
 3.989 
 
 4.007 
 
 4.025 
 
 4.043 
 
 4.060 
 
 4.078 
 
 6 
 
 4.094 
 
 4.111 
 
 4.127 
 
 4.143 
 
 4.159 
 
 4.174 
 
 4.190 
 
 4.205 
 
 4.220 
 
 4.234 
 
 7 
 
 4.248 
 
 4.263 
 
 4.277 
 
 4.290 
 
 4.304 
 
 4.317 
 
 4.331 
 
 4.344 
 
 4.357 
 
 4.369 
 
 8 
 
 4.382 
 
 4.394 
 
 4.407 
 
 4.419 
 
 4.431 
 
 4.443 
 
 4.454 
 
 4.466 
 
 4.477 
 
 4.489 
 
 9 
 
 4.500 
 
 4.511 
 
 4.522 
 
 4.533 
 
 4.543 
 
 4.554 
 
 4.564 
 
 4.575 
 
 4.585 
 
 4.595 
 
 10 
 
 4.605 
 
 4.615 
 
 4.625 
 
 4.635 
 
 4.644 
 
 4.654 
 
 4.663 
 
 4.673 
 
 4.682 
 
 4.691 
 
 The Values in Circular Measure of Angles which are given In 
 Degrees and Minutes. 
 
 1' 
 
 0.0003 
 
 9' 
 
 0.0026 
 
 3° 
 
 0.0524 
 
 20° 
 
 0.3491 
 
 100° 
 
 1.7453 
 
 V 
 
 0.0006 
 
 10' 
 
 0.0029 
 
 40 
 
 0.0698 
 
 30° 
 
 0.5236 
 
 110° 
 
 1.9199 
 
 V 
 
 0.0009 
 
 20' 
 
 0.0058 
 
 5° 
 
 0.0873 
 
 40° 
 
 0.6981 
 
 120° 
 
 2.0944 
 
 4' 
 
 0.0012 
 
 30' 
 
 0.0087 
 
 (P 
 
 0.1047 
 
 50° 
 
 0.8727 
 
 130° 
 
 2.2689 
 
 5' 
 
 0.0015 
 
 40' 
 
 0.01 16 
 
 70 
 
 0.1222 
 
 60° 
 
 1.0472 
 
 140° 
 
 2.4435 
 
 6' 
 
 0.0017 
 
 50' 
 
 0.0145 
 
 8° 
 
 0.1396 
 
 70° 
 
 1.2217 
 
 150° 
 
 2.6180 
 
 V 
 
 0.0020 
 
 1° 
 
 0.0175 
 
 90 
 
 0.1571 
 
 80° 
 
 1.3963 
 
 160° 
 
 2.7925 
 
 8' 
 
 0.0023 
 
 2° 
 
 0.0349 
 
 10° 
 
 0.1745 
 
 90° 
 
 1.5708 
 
 170° 
 
 2.9671 
 
30 
 
 TABLES. 
 
 NATURAL TRIGONOMETRIC FUNCTIONS. 
 
 Angle. 
 
 Sin. 
 
 Csc. 
 
 Tan. 
 
 Ctn. 
 
 Sec. 
 
 Cos. 
 
 
 0° 
 
 0.000 
 
 00 
 
 0.000 
 
 00 
 
 1.000 
 
 1.000 
 
 90° 
 
 1 
 
 0.017 
 
 57.30 
 
 0.017 
 
 57.29 
 
 1.000 
 
 1.000 
 
 89 
 
 2 
 
 0.035 
 
 28.65 
 
 0.035 
 
 28.64 
 
 1.001 
 
 0.999 
 
 88 
 
 3 
 
 0.052 
 
 19.11 
 
 0.052 
 
 19.08 
 
 1.001 
 
 0.999 
 
 87 
 
 4 
 
 0.070 
 
 14.34 
 
 0.070 
 
 14.30 
 
 1.002 
 
 0.998 
 
 86 
 
 5° 
 
 0.087 
 
 11.47 
 
 0.087 
 
 11.43 
 
 1.004 
 
 0.996 
 
 85° 
 
 6 
 
 0.105 
 
 9.567 
 
 0.105 
 
 9.514 
 
 1.006 
 
 0.995 
 
 84 
 
 7 
 
 0.122 
 
 8.206 
 
 0.123 
 
 8.144 
 
 1.008 
 
 0.993 
 
 83 
 
 8 
 
 0.139 
 
 7.185 
 
 0.141 
 
 7.115 
 
 1.010 
 
 0.990 
 
 82 
 
 9 
 
 0.156 
 
 6.392 
 
 0.158 
 
 6.314 
 
 1.012 
 
 0.988 
 
 81 
 
 10° 
 
 0.174 
 
 5.759 
 
 0.176 
 
 5.671 
 
 1.015 
 
 0.985 
 
 80° 
 
 11 
 
 0.191 
 
 5.241 
 
 0.194 
 
 5.145 
 
 1.019 
 
 0.982 
 
 1 79 
 
 12 
 
 0.208 
 
 4.810 
 
 0.213 
 
 4.705 
 
 1.022 
 
 0.978 
 
 78 
 
 13 
 
 0.225 
 
 4.445 
 
 0.231 
 
 4.331 
 
 1.026 
 
 0.974 
 
 77 
 
 14 
 
 0.242 
 
 4.134 
 
 0.249 
 
 4.011 
 
 1.031 
 
 0.970 
 
 76 
 
 15° 
 
 0.259 
 
 3.864 
 
 0.268 
 
 3.732 
 
 1.035 
 
 0.966 
 
 75° 
 
 16 
 
 0.276 
 
 3.628 
 
 0.287 
 
 3.487 
 
 1.040 
 
 0.961 
 
 74 
 
 17 
 
 0.292 
 
 3.420 
 
 0.306 
 
 3.271 
 
 1.046 
 
 0.956 
 
 73 
 
 18 
 
 0.309 
 
 3.236 
 
 0.325 
 
 3.078 
 
 1.051 
 
 0.951 
 
 72 
 
 19 
 
 0.326 
 
 3.072 
 
 0.344 
 
 2.904 
 
 1.058 
 
 0.946 
 
 71 
 
 20° 
 
 0.342 
 
 2.924 
 
 0.364 
 
 2.747 
 
 1.064 
 
 0.940 
 
 70° 
 
 21 
 
 0.358 
 
 2.790 
 
 0.384 
 
 2.605 
 
 1.071 
 
 0.934 
 
 69 
 
 22 
 
 0.375 
 
 *2.669 
 
 0.404 
 
 2.475 
 
 1.079 
 
 0.927 
 
 68 
 
 23 
 
 0.391 
 
 2.559 
 
 0.424 
 
 2.356 
 
 1.086 
 
 0.921 
 
 67 
 
 24 
 
 0.407 
 
 2.459 
 
 0.445 
 
 2.246 
 
 1.095 
 
 0.914 
 
 66 
 
 25- 
 
 0.423 
 
 2.366 
 
 0.466 
 
 2.145 
 
 1.103 
 
 0.906 
 
 65° 
 
 26 
 
 0.438 
 
 2.281 
 
 0.488 
 
 2.050 
 
 1.113 
 
 0.899 
 
 64 
 
 27 
 
 0.454 
 
 2.203 
 
 0.530 
 
 1.963 
 
 1.122 
 
 0.891 
 
 63 
 
 28 
 
 0.469 
 
 2.130 
 
 0.532 
 
 1.881 
 
 1.133 
 
 0.883 
 
 62 
 
 29 
 
 0.485 
 
 2.063 
 
 0.554 
 
 1.804 
 
 1.143 
 
 0.875 
 
 61 
 
 30° 
 
 0.500 
 
 2.000 
 
 0.577 
 
 1.732 
 
 1.155 
 
 0.866 
 
 60° 
 
 31 
 
 0.515 
 
 1.942 
 
 0.601 
 
 1.664 
 
 1.167 
 
 0.857 
 
 59 
 
 32 
 
 0.530 
 
 1.887 
 
 0.625 
 
 1.600 
 
 1.179 
 
 0.848 
 
 58 
 
 33 
 
 0.545 
 
 1.836 
 
 0.649 
 
 1.54Q 
 
 1.192 
 
 0.839 
 
 57 
 
 34 
 
 0.559 
 
 1.788 
 
 0.675 
 
 1.483 
 
 1.206 
 
 0.829 
 
 56 
 
 35° 
 
 0.574 
 
 1.743 
 
 0.700 
 
 1.428 
 
 1.221 
 
 0.819 
 
 55° 
 
 36 
 
 0.588 
 
 1.701 
 
 0.727 
 
 1.376 
 
 1.236 
 
 0.809 
 
 54 
 
 37 
 
 0.602 
 
 1.662 
 
 0.754 
 
 1.327 
 
 1.252 
 
 0.799 
 
 53 
 
 38 
 
 0.616 
 
 1.624 
 
 0.781 
 
 1.280 
 
 1.269 
 
 0.788 
 
 52 
 
 39 
 
 0.629 
 
 1.589 
 
 0.810 
 
 1.235 
 
 1.287 
 
 0.777 
 
 51 
 
 40° 
 
 0.643 
 
 1.556 
 
 0.839 
 
 1.192 
 
 1.305 
 
 0.766 
 
 50° 
 
 41 
 
 0656 
 
 1.524 
 
 0.869 
 
 1.150 
 
 1.325 
 
 0.755 
 
 49 
 
 42 
 
 0.669 
 
 1.494 
 
 0.900 
 
 1.111 
 
 1.346 
 
 0.743 
 
 48 
 
 43 
 
 0.682 
 
 1.466 
 
 0.933 
 
 1.072 
 
 1.367 
 
 0.731 
 
 47 
 
 44 
 
 0.695 
 
 1.440 
 
 0.966 
 
 1.036 
 
 1.390 
 
 0.719 
 
 46 
 
 45° 
 
 0.707 
 
 1.414 
 
 1.000 
 
 1.000 
 
 1.414 
 
 0.707 
 
 45° 
 
 
 Cos. 
 
 Sec. 
 
 Ctn. 
 
 Tan. 
 
 Oec. 
 
 Sin. 
 
 Anglo. 
 
TABLES. 
 
 31 
 
 Values of the Complete Elliptic Integrals, K and E, for Different 
 Values of the Modulus, k. 
 
 sin-iifc 
 
 K 
 
 E 
 
 sin-iA; 
 
 K 
 
 E 
 
 sin-U- 
 
 K 
 
 E 
 
 0° 
 
 1.5708 
 
 1.5708 
 
 30^^ 
 
 ].6858 
 
 1.4675 
 
 60° 
 
 2.1565 
 
 1.2111 
 
 1° 
 
 1.5709 
 
 1.5707 
 
 31° 
 
 1.6941 
 
 ].4608 
 
 61° 
 
 2.1842 
 
 1.2015 
 
 2° 
 
 1.5713 
 
 1.5703 
 
 32° 
 
 1.7028 
 
 1.4539 
 
 62° 
 
 2.2132 
 
 1.1920 
 
 3° 
 
 1.5719 
 
 1.5697 
 
 33° 
 
 1.7119 
 
 1.4469 
 
 63° 
 
 2.2435 
 
 1.1826 
 
 40 
 
 1.5727 
 
 1.5689 
 
 34° 
 
 1.7214 
 
 1.4397 
 
 64° 
 
 2.2754 
 
 1.1732 
 
 50 
 
 1.5738 
 
 1.5678 
 
 35° 
 
 1.7312 
 
 1.4223 
 
 65° 
 
 2.3088 
 
 1.1638 
 
 6° 
 
 1.5711 
 
 1.5665 
 
 36° 
 
 1.7415 
 
 1.4248 
 
 66° 
 
 2.3439 
 
 1.1545 
 
 70 
 
 1.5767 
 
 1.5649 
 
 37° 
 
 1.7522 
 
 1.4171 
 
 67° 
 
 2.3809 
 
 1.1453 
 
 8° 
 
 1.5785 
 
 1.5632 
 
 38° 
 
 1.7633 
 
 1.4092 
 
 68° 
 
 2.4198 
 
 1.1362 
 
 90 
 
 1.5805 
 
 1.5611 
 
 39° 
 
 1.7748 
 
 1.4013 
 
 69° 
 
 2.4610 
 
 1.1272 
 
 10° 
 
 1.5828 
 
 1.5589 
 
 40° 
 
 1.7868 
 
 1.3931 
 
 70° 
 
 2.5046 
 
 1.1184 
 
 11° 
 
 1.5854 
 
 1.5564 
 
 41° 
 
 1.7992 
 
 1.3849 
 
 71° 
 
 2.5507 
 
 1.1096 
 
 12° 
 
 1.5882 
 
 1.5537 
 
 42° 
 
 1.8122 
 
 1.3765 
 
 72° 
 
 2.5998 
 
 1.1011 
 
 13° 
 
 1.5913 
 
 1.5507 
 
 43° 
 
 1.8256 
 
 1.3680 
 
 73° 
 
 2.6521 
 
 1.0927 
 
 14° 
 
 1.5946 
 
 1.5476 
 
 44° 
 
 1.8396 
 
 1.3594 
 
 74° 
 
 2.7081 
 
 1.0844 
 
 15° 
 
 1.5981 
 
 1.5442 
 
 45° 
 
 1.8541 
 
 1.3506 
 
 75° 
 
 2.7681 
 
 1.0764 
 
 16° 
 
 1.6020 
 
 1.5405 
 
 46° 
 
 1.8691 
 
 1.3418 
 
 76° 
 
 2.8327 
 
 1.0686 
 
 17° 
 
 1.6061 
 
 1.5367 
 
 47° 
 
 1.8848 
 
 1.3329 
 
 77° 
 
 2.9026 
 
 1.0611 
 
 18° 
 
 1.6105 
 
 1.5326 
 
 48° 
 
 1.9011 
 
 1.3238 
 
 78° 
 
 2.9786 
 
 1.0538 
 
 19° 
 
 1.6151 
 
 1.5283 
 
 49° 
 
 1.9180 
 
 1.3147 
 
 79° 
 
 3.0617 
 
 1.0468 
 
 20° 
 
 1.6200 
 
 1.5238 
 
 50° 
 
 1.9356 
 
 1.3055 
 
 80° 
 
 3.1534 
 
 1.0401 
 
 21° 
 
 1.6252 
 
 1.5191 
 
 51° 
 
 1.9539 
 
 1.2963 
 
 81° 
 
 3.2553 
 
 1.0338 
 
 22° 
 
 1.6307 
 
 1.5141 
 
 52° 
 
 1.9729 
 
 1.2870 
 
 82° 
 
 3.3699 
 
 1.0278 
 
 23° 
 
 1.6365 
 
 1.5090 
 
 53° 
 
 1.9927 
 
 1.2776 
 
 83° 
 
 3.5004 
 
 1.0223 
 
 24° 
 
 1.6426 
 
 1.5037 
 
 54° 
 
 2.0133 
 
 1.2681 
 
 84° 
 
 3.6519 
 
 1.0172 
 
 25° 
 
 1.6490 
 
 ].4981 
 
 55° 
 
 2.0347 
 
 1.2587 
 
 85° 
 
 3.8317 
 
 1.0127 
 
 26° 
 
 1.6557 
 
 1.4924 
 
 56° 
 
 2.0571 
 
 1.2492 
 
 86° 
 
 4.0528 
 
 1.0086 
 
 27° 
 
 1.6627 
 
 1.4864 
 
 57° 
 
 2.0804 
 
 1.2397 
 
 87° 
 
 4.3387 
 
 1.0053 
 
 28° 
 
 1.6701 
 
 1.4803 
 
 58° 
 
 2.1047 
 
 1.2301 
 
 88° 
 
 4.7427 ! 1.0026 
 
 29° 
 
 1.6777 
 
 1.4740 
 
 1 59° 
 
 1 
 
 2.1300 
 
 1.2206 
 
 89° 
 
 5.4349 1 1.0008 
 
TABLES. 
 
 The Common Logarithms ot r(«) ^o'' Values of n between 1 and 2. 
 
 n 
 
 i 
 
 n 
 
 s 
 
 o 
 
 i 
 
 n 
 
 Is 
 
 n 
 
 s 
 
 S 
 
 ■ 
 n 
 
 1 
 
 
 
 
 
 
 
 _ 
 
 
 _ 
 
 1.01 
 
 1.9975 
 
 1.21 
 
 1.9617 
 
 1.41 
 
 1.9478 
 
 1.61 
 
 1.9517 
 
 1.81 
 
 1.9704 
 
 1.02 
 
 1.9951 
 
 1.22 
 
 r.9605 
 
 1.42 
 
 1.9476 
 
 1.62 
 
 f.9523 
 
 1.82 
 
 1.9717 
 
 1.03 
 
 1.9928 
 
 1.23 
 
 r.9594 
 
 1.43 
 
 r.9475 
 
 1.63 
 
 1.9529 
 
 1.83 
 
 1.9730 
 
 1.04 
 
 1.9905 
 
 1.24 
 
 1.9583 
 
 1.44 
 
 1.9473 
 
 1.64 
 
 1.9536 
 
 1.84 
 
 1.9743 
 
 1.05 
 
 1.9883 
 
 1.25 
 
 1.9573 
 
 1.45 
 
 1.9473 
 
 1.65 
 
 1.9543 
 
 1.85 
 
 1.9757 
 
 V06 
 
 L9862 
 
 1.26 
 
 1.9564 
 
 1.46 
 
 1.9472 
 
 1.66 
 
 1.9550 
 
 1.86 
 
 1.9771 
 
 1.07 
 
 r.9841 
 
 1.27 
 
 1.9554 
 
 1.47 
 
 1.9473 
 
 1.67 
 
 1.9558 
 
 1.87 
 
 1.9786 
 
 1.08 
 
 1.9821 
 
 1.28 
 
 1.9546 
 
 1.48 
 
 1.9473 
 
 1.68 
 
 r.9566 
 
 1.88 
 
 1.9800 
 
 1.09 
 
 1.9802 
 
 1.29 
 
 r.9538 
 
 1.49 
 
 1.9474 
 
 1.69 
 
 1.9575 
 
 1.89 
 
 1.9815 
 
 1.10 
 
 1.9783 
 
 1.30 
 
 1.9530 
 
 1.50 
 
 1.9475 
 
 1.70 
 
 r.9584 
 
 1.90 
 
 1.9831 
 
 1.11 
 
 1.9765 
 
 1.31 
 
 1.9523 
 
 1.51 
 
 1.9477 
 
 1.71 
 
 r.9593 
 
 1.91 
 
 1.9846 
 
 1.12 
 
 1.9748 
 
 1.32 
 
 1.9516 
 
 1.52 
 
 1.9479 
 
 1.72 
 
 1.9603 
 
 1.92 
 
 1.9862 
 
 1.13 
 
 1.9731 
 
 1.33 
 
 1.9510 
 
 1.53 
 
 r.9482 
 
 1.73 
 
 1.9613 
 
 1.93 
 
 1.9878 
 
 1.14 
 
 1.9715 
 
 1.34 
 
 f.9505 
 
 1.54 
 
 1.9485 
 
 1.74 
 
 1.9623 
 
 1.94 
 
 1.9895 
 
 1.15 
 
 1.9699 
 
 1.35 
 
 1.9500 
 
 1.55 
 
 1.9488 
 
 1.75 
 
 1.9633 
 
 1.95 
 
 19912 
 
 1.16 
 
 1.9684 
 
 1.36 
 
 1.9495 
 
 1.56 
 
 1.9492 
 
 1.76 
 
 r.9644 
 
 1.96 
 
 1.9929 
 
 1.17 
 
 1.9669 
 
 1.37 
 
 1.9491 
 
 1.57 
 
 1.9496 
 
 1.77 
 
 1.9656 
 
 1.97 
 
 1.9946 
 
 1.18 
 
 1.9655 
 
 1.38 
 
 1.9487 
 
 1.58 
 
 1.9501 
 
 1.78 
 
 1.9667 
 
 1.98 
 
 1.9964 
 
 1.19 
 
 1.9642 
 
 1.39 
 
 1.9483 
 
 1.59 
 
 1.9506 
 
 1.79 
 
 1.9679 
 
 1.99 
 
 1.9982 
 
 1.20 
 
 1 
 
 19629 
 
 1.40 
 
 • 
 
 1.9481 
 
 1.60 
 
 1.9511 
 
 1 
 
 1.80 
 
 1.9691 
 
 2.00 
 
 0.0000 
 
"'€!?; 
 
 RETURN TO Dpt £^^ USE m 
 
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 LOAN DEPT 
 
 ^-- ^__»jmmed.a,a recal,. 
 
 r I. •*^°"»' library 
 
^11 O i 
 
 7s- -t is~ 
 
 U. C. BERKELEY LIBRARIES 
 
 CDS7^EDfiED 
 
 86B425 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
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