"
,..
x·-
TABLE
,
oC
IN'l'EGRALS
f Va 1 x sm -a + C.
1
x + c.'.
14.
J +.1 dx -tan1
1
x
IS. ' v'
dx = -sec- + C.
J
a
a
a
16. J!(x)g'(x) d.t I(x)g(x) ...:. Jg(x)/'(x) dx.
1.
, = _ r.
l. ln v'
vi ax + b-' Vb
Vbl
1 7J. _ r--:-;dX
+ C, (0) 0). '
vb
ax+b+·.b
18. J V'I
b dx =='._2
;--;- tan~l ~
ax + b + C, (b -< 0).
x ax +
-b-b,
'.JJ.
13.
2 -
2
"!II:
=
a2
X2
=
X
-1
•
1-
'a
\'
,a
1
1.
I
,
tf<x)
± g(x)] dx
=
flOe) ax ± f g(x) dx.
Jtf{x) dx = ~ Jf(x) dx. 3; Jf(t{x}jgj(x)dx J!(ti)dulu..
4. '
j":it"
L x..+ + C' (11 -1).
. 'dt »+1
2.
=
'(Z) 1
:!i:
¢
IS.
! sili xax;" ~cos x+ c.
7.
! b~)'s,~ dx == sin x +C.
8.
j sect x .dx
=
tan, x
12. f'i!1:llx = e"z
=
+ ,t.·
-
2
=
I'
xv~x+b
V
f'
. 1
t v ' aX + b
. dx == _
Vax + b
x-- 1
b(n - 1)
(2n- 3)0
(2n - 2)b
1 . . .. ".
f )("--1"(1)£ + b dx,:(r;a
¢
I).
.
J V'I dx.
f Vax. + b dx=2Yax+b+b
x ax+b
.1
J . 1d x -In
1 Ix",:,
2
- -al + C.
2a x +
20. ,"
y
+ C.
f esc! x die == -cot x + c.
10. f Sec x tan x dx
Sec x +
11. f esc
xdx
-esc x -+ t, '
X,cot
X2 -
X-
9.
=
X
19.
5. .f' :t.
lilt .l:bilxl + c;
(
C; .
X2 -
.
=
a'l.
a
1"
I . lex + d/ ' ','
", .
+ d/x lie _ odin ax + b + C. (be - ad
J(ax + b)(cx
.
23. J(ax +
+d) dx ' ,be.-1 ad{£
In/ax
+ bl';" "~.' .' ~n l<:x +,,(11'. f1 ",
b~
'. ex,
a
, =
22.
¢Q).
.
',~
=
4- C. (~,.,.. i(l(t,#,·O~.
4J
1
2. (ax
+ bWcx +
dx
d)
I{
I
,e ,11cx +/d/l­
=bc - ad ax +b -+ be - -ad n~aK+~j1 .
+C. ~t;b~ -1;1(1;;4 :O~.
.
25.
f
.
(ax
x·
{_b
.--k-~ _1
+ bl'lCX+ d ) i /
In"" +
:r} +
+ be -:.. ad I~x + ,
..
C. (be _ at/;4 '0).
I
'­
Page ) Table of Integrals
f V, x~ ± a'dx
26.
\
,
I
,28. I
27.
x
i='
'2 V x
±a2
1"
.
at
X2"'; Xl
±
a2 dx
= ~ (2x ± ( V x ± a
2
)
2
2
2
I V x' ± a dx
~
,
I
'
2
~
-
gIn Ix + V x 2 ± a'll
'
V x' +- a2 + -2
In'Ix· ·+,,V' X2 +- a'l + C•
2
± a'Jl/2
x(x2
3 I x' V X2 ± a2 ·dx.
-
±x "
1,
I x ±a,
I (x ;2a-' dx == ...; -:x± + In Ix + ~ x'. ± a2+ C. "I
1
33.dx
+" Vx'+-- a + C. (2:
32.
dx =
2\312
2
~
V X2 ± a'' + c. ''1/4
2:
02
X2
V x' ± a
x'
34.
I~
a dx
,
x
2
, I' vx
35.
36.
I
2
2
± a + In Ix + V X2 ± a 21+ c. ± ~2dx -- VX2 +
a2 +
x '--:
02
I
I., .dx.
a~
x ...; x 2: +
~
~ Inla + VX +0 \+ C.
dx =,_
at
'
a"
x
'
-'x2 dx ;.;:
,2,
2
Vat --
'
2
2
47.
I
=
X2 -
a"
f Va' -- dx == .... 4
,x
x' --:....
39.
dx
Va2 .::.. X2
f Va2- x'
2,
38. .
X2
(a
X2
2
,
= -- ....,
'a'll
-- X2)11'
-2
+"4 ...; a
a2 x
~,- sin-I -:2
a
1­
(X2
+a2 )" dx
'.
I
{
- 2(n -I)ot (x'
x
:.
+ a2)..-1 + (~-- 3)
,
--
x 2'1'2
-- X2
,
dx. 1
)"-1 dx j , (:>I ;t! 1).
I x sin x dx == sin x -- x cos x + c.
49. I
sin x dx --x~ cos x + ;'.~-I sin x -- n(n -- I) I
sin dx.
50. I x cOs x dx "" cos x + x sin x + c.
51. f x" cos x dx, x" sin x +
cos x·- n(n ,;;,. I x·- cos x dx.
52. I sin'"
x dx
_'_1_ [._ sin...
+
I) I sin'"-! cos~ dx]
+n,
'
=
X"
X"-2
1)
1lX"-1
x COS"+! x
(m -
={
'[-'_1_ {sin....H x cos,.-I x
'
m+n + c.
dX = ~ (a 2 -- 'x'l}'!~,I 4
:,a I·:"\.;/ a1 --~; dx. '
.4
I (x' +Ia2'
48.
2
40. '/', (a 2
c.
X2\'+
1
--
-1
x
·
2
Xi
2 -
."r;.
2
COS"
~ Va' __ )(2 + a sin-l~ + C. 2
02
"I
=
2
2
V X2 +
37;J>,·v'
= - v'x
2
1
x
a'x'
--
X2
I x' Va x' dx=-- atx +c. .
45. I va -x d.'(
Va' _
alnla + Va'.x
",
x
46. I V a
x! dx:;' ~ ...; at ..:... x' ~ sin- ~ + c. x'
x
a
~
'
2:
=
2
1
"
+ C.
2
-./
31.
• -
= '_
2 -
= -
30. I(X 2 ± 02Jll 2 dx =
,
2
43.
'"
~.
'
2 -- x'
4
•
29.
Ix
'=
2 -- x2}311
X
,-
x2 ±
))
)
. ax
+ c.
,I (a
a2 V. a
42. f ( , :.2 2Jl/~ ax::= V
sin-I ~ + c.
,a
x
a--x-'
a
J ...; 1 dx !a lu,la +Vax x'l+ c.
,. x a
Ix + ...; x 2 ± a 21+ C. dx == In
')
41. '
± "2 In Ix + Vx' ± a21 + C. '
V
,
~
2
't
_ 53.
,
I sm". x dx
"1.
sm,,-l x cos x
n,
= - -
+ (n --,
I-
I)
x
I sin'"
x C05"-:
(m
.
1
+ n--:---sm"-~ X dx, (n 2:: 2).
n
.Y.
.~dxJ.
1:'= 0).
Pager;
.(
1.-
Tabl~
Integrals
x
· d 1 .
54. I 5m x x = - "2 sui x cos x + '2 + C
69.
2
cos.. x dx = -l
sin x'COS,,-I x
2
= -
n
72. /
3
SlO X
73.
=
2
x dx = 61.
x
x dx = 2
/
:x!'
Jxe
u
dx =
78.
I )e"" + C.
-l
f
r"'
In"·' Ixl
ax)."
~
-1).
Ixl dx.
1 )
X..+l ( .
+ c, (n/'F-
-1).
'
x In Ixl dx "" In lIn !xll + C.
Jsin": xdx
l
=
xsin-1x+ ~ + C.
1 {
,
1
X"H sin-1
JX",sin-1
+ 1. •
v'1 80. f tan-I xdx = x tan-I x - ~ In (x + 1) + C.
79.
xdx = - n
X"+1
x -
} ' .,
dx , (n ~ -1).
X2
' ,
2
~
2).
x
~2 (ax.-
n
In Ixl dx = n + 1 In Ixl - n + 1 + C, (n ~ -1).
1
2
(n
~ 1(x-+' 1n"lxl -
75.
,
'1 {
'''+1}'
81. / X" tan-1 xdx = - - X,,+I tan- 1 x -, / _x__ dx , (n ~ -1).
.
xl
dx
- m
+ C.
,J In"/xl dx ~ x In" Ixl'- n JIn..
77. /
f cscxdx == In Icscx - cot + C.
1 { -CSC"'-2 cot X + (n - 2) f.C~"'-2
' }
67. 1csc" x , = ;;-=}
X dx •
66.
68.
r"' 1n"lxl <Ix
In Ixl - x
In" Ixl
t . '
76. /, - x - dx = n + 1 In,,+1 Ixl
=
(n
= x
74.
1
2
'
J
=
=
n Ixl dx
x dx,
xl
dx ',;" -
1
.
b2 (a cos bx + b sin bx)e"" + C.
(m
x - x
x dx
ar
I'
= a2 +
71. / e"" cos bx dx
Jtanxdx In sec xl + C.
59. 1tan
tan
+ C.
60. f tan"
== _1_ tan",-l ~ Jtan,d
(n ~ 2).
n- 1
f cot
In ;sin + C.
62. f cot xdx
-cot x· - x + C.
63. 1cot" x
n ~ 1 cot..- x - f cot,,-2 x dx; (n 2': 2).
64. f sec x dx In Isec x + tan xl + C.
65. f sec" x dx n ~ I {sec..- x tan x + (n - 2) f sec"':' Xdx}.
58.
111 x..-1e ' dx.
x" = -;; e'"'' - ~
70. / eo" sin bx dx = -;~b2 (a sin bx - b cos bx)e°;t: + C.
, . a +
+ n-IJ'
- - COS,,-2 X dx, (n ~ 2).
1
n' 56. Jcos' x dx = ~ sin x cos x + ~ + c.
. x cos x + 8
x+c.
. x cos-·x dx 4:1 .cos1
57. 1sm
x +8 sm
55.
.
/ X"e"" dx
5 2).
82~
n+I f .sec- x dx := x sec-I x 1
~+I'
In Ix + Vx 2
-
11 + C;