A short table of integrals

^ 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

"^ TO DBSK FROM WHICH BORKOwS

LOAN DEPT

^-- ^__»jmmed.a,a recal,.

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