integrals

1. Table of Basic Integrals
Basic Forms
(1)
Z
xndx =
1
n + 1
xn+1; n6= 1
(2)
Z
1
x
dx = ln jxj
(3)
Z
udv = uv
Z
vdu
(4)
Z
1
ax + b
dx =
1
a
ln jax + bj
Integrals of Rational Functions
(5)
Z
1
(x + a)2 dx =
1
x + a
(6)
Z
(x + a)ndx =
(x + a)n+1
n + 1
; n6= 1
(7)
Z
x(x + a)ndx =
(x + a)n+1((n + 1)x a)
(n + 1)(n + 2)
(8)
Z
1
1 + x2 dx = tan1 x
(9)
Z
1
a2 + x2 dx =
1
a
tan1 x
a
1

2. (10)
Z
x
a2 + x2 dx =
1
2
ln ja2 + x2j
(11)
Z
x2
x
dx = x a tan1 a2 + x2 a
(12)
Z
x3
a2 + x2 dx =
1
2
x2
1
2
a2 ln ja2 + x2j
(13)
Z
1
ax2 + bx + c
dx =
2
p
4ac b2
tan1 2ax + b
p
4ac b2
(14)
Z
1
(x + a)(x + b)
dx =
1
b a
ln
a + x
b + x
; a6= b
(15)
Z
x
(x + a)2 dx =
a
a + x
+ ln ja + xj
(16)
Z
x
ax2 + bx + c
dx =
1
2a
ln jax2+bx+cj
b
p
4ac b2
a
tan1 2ax + b
p
4ac b2
Integrals with Roots
(17)
Z
p
x a dx =
2
3
(x a)3=2
(18)
Z
1
p
x a
p
x a
dx = 2
(19)
Z
1
p
a x
p
a x
dx = 2
2

3. (20)
Z
x
p
x a dx =
8
:
2a
3 (x a)3=2 + 2
5 (x a)5=2 ; or
2
3x(x a)3=2 4
15 (x a)5=2; or
2
15(2a + 3x)(x a)3=2
(21)
Z p
ax + b dx =
2b
3a
+
2x
3
p
ax + b
(22)
Z
(ax + b)3=2 dx =
2
5a
(ax + b)5=2
(23)
Z
x
p
x a
dx =
2
3
p
x a
(x 2a)
(24)
Z r
x
a x
p
x(a x) a tan1
dx =
p
x(a x)
x a
(25)
Z r
x
a + x
dx =
p
x(a + x) a ln
p
x +
p
x + a
(26)
Z
x
p
ax + b dx =
2
15a2 (2b2 + abx + 3a2x2)
p
ax + b
(Z27)p
x(ax + b) dx =
1
4a3=2
h
(2ax + b)
p
ax(ax + b) b2 ln

6. p
x +
a

9. p
a(ax + b)
i
(Z28)p
x3(ax + b) dx =
b
12a
b2
8a2x
+
x
3
p
x3(ax + b)+
b3
8a5=2 ln

12. p
x +
a
p
a(ax + b)

15. (29)
Z p
x2 a2 dx =
1
2
x
p
x2 a2
1
2
a2 ln

18. x +
p
x2 a2

21. 3

22. (30)
Z p
a2 x2 dx =
1
2
x
p
a2 x2 +
1
2
a2 tan1 x
p
a2 x2
(31)
Z
x
p
x2 a2 dx =
1
3
x2 a23=2
(32)
Z
1
p
x2 a2
dx = ln

25. x +
p
x2 a2

28. (33)
Z
1
p
a2 x2
dx = sin1 x
a
(34)
Z
x
p
x2 a2
dx =
p
x2 a2
(35)
Z
x
p
a2 x2
p
a2 x2
dx =
(36)
Z
x2
p
x2 a2
dx =
1
2
x
p
x2 a2
1
2
a2 ln

31. x +
p
x2 a2

34. (Z37)p
ax2 + bx + c dx =
b + 2ax
4a
p
ax2 + bx + c+
4ac b2
8a3=2 ln

37. 2ax + b + 2

40. p
a(ax2 + bx+c)
Z
x
p
ax2 + bx + c dx =
1
48a5=2
2
p
ax2 + bx + c
p
a
3b2 + 2abx + 8a(c + ax2)
+3(b3 4abc) ln

43. b + 2ax + 2
p
ax2 + bx + c
p
a

46. (38)
4

47. (39)
Z
1
p
ax2 + bx + c
dx =
1
p
a
ln

50. 2ax + b + 2
p
a(ax2 + bx + c)

53. (Z40)
x
p
ax2 + bx + c
dx =
1
a
p
ax2 + bx + c
b
2a3=2 ln

56. 2ax + b + 2
p
a(ax2 + bx + c)

59. (41)
Z
dx
(a2 + x2)3=2 =
x
a2
p
a2 + x2
Integrals with Logarithms
(42)
Z
ln ax dx = x ln ax x
(43)
Z
x ln x dx =
1
2
x2 ln x
x2
4
(44)
Z
x2 ln x dx =
1
3
x3 ln x
x3
9
(45)
Z
xn ln x dx = xn+1
ln x
n + 1
1
(n + 1)2
; n6= 1
(46)
Z
ln ax
x
dx =
1
2
(ln ax)2
(47)
Z
ln x
x2 dx =
1
x
ln x
x
5

60. (48)
Z
ln(ax + b) dx =
x +
b
a
ln(ax + b) x; a6= 0
(49)
Z
ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan1 x
a
2x
(50)
Z
ln(x2 a2) dx = x ln(x2 a2) + a ln
x + a
x a
2x
(Z51)
ln
ax2 + bx + c
dx =
1
a
p
4ac b2 tan1 2ax + b
p
4ac b2
2x+
b
2a
+ x
ln
ax2 + bx + c
(52)
Z
x ln(ax + b) dx =
bx
2a
1
4
x2 +
1
2
x2
b2
a2
ln(ax + b)
(53)
Z
x ln
a2 b2x2
dx =
1
2
x2 +
1
2
x2
a2
b2
ln
a2 b2x2
(54)
Z
(ln x)2 dx = 2x 2x ln x + x(ln x)2
(55)
Z
(ln x)3 dx = 6x + x(ln x)3 3x(ln x)2 + 6x ln x
(56)
Z
x(ln x)2 dx =
x2
4
+
1
2
x2(ln x)2
1
2
x2 ln x
(57)
Z
x2(ln x)2 dx =
2x3
27
+
1
3
x3(ln x)2
2
9
x3 ln x
6

61. Integrals with Exponentials
(58)
Z
eax dx =
1
a
eax
(5Z9)
p
xeax dx =
1
a
p
xeax +
p
i
erf
2a3=2
i
p
ax
; where erf(x) =
2
p
Z x
0
et2
dt
(60)
Z
xex dx = (x 1)ex
(61)
Z
xeax dx =
x
a
1
a2
eax
(62)
Z
x2ex dx =
x2 2x + 2
ex
(63)
Z
x2eax dx =
x2
a
2x
a2 +
2
a3
eax
(64)
Z
x3ex dx =
x3 3x2 + 6x 6
ex
(65)
Z
xneax dx =
xneax
a
n
a
Z
xn1eaxdx
(66)
Z
xneax dx =
(1)n
an+1 [1 + n;ax]; where (a; x) =
Z 1
x
ta1et dt
(67)
Z
eax2
dx =
p
i
2
p
a
erf
ix
p
a
7

62. (68)
Z
eax2
dx =
p
2
p
a
erf
x
p
a
(69)
Z
xeax2
dx =
1
2a
eax2
(70)
Z
x2eax2
dx =
1
4
r
a3 erf(x
p
a)
x
2a
eax2
Integrals with Trigonometric Functions
(71)
Z
sin ax dx =
1
a
cos ax
(72)
Z
sin2 ax dx =
x
2
sin 2ax
4a
(73)
Z
sin3 ax dx =
3 cos ax
4a
+
cos 3ax
12a
(74)
Z
sinn ax dx =
1
a
cos ax 2F1
1
2
;
1 n
2
;
3
2
; cos2 ax
(75)
Z
cos ax dx =
1
a
sin ax
(76)
Z
cos2 ax dx =
x
2
+
sin 2ax
4a
(77)
Z
cos3 axdx =
3 sin ax
4a
+
sin 3ax
12a
8

63. (78)
Z
cosp axdx =
1
a(1 + p)
cos1+p ax 2F1
1 + p
2
;
1
2
;
3 + p
2
; cos2 ax
(79)
Z
cos x sin x dx =
1
2
sin2 x + c1 =
1
2
cos2 x + c2 =
1
4
cos 2x + c3
(80)
Z
cos ax sin bx dx =
cos[(a b)x]
2(a b)
cos[(a + b)x]
2(a + b)
; a6= b
(81)
Z
sin2 ax cos bx dx =
sin[(2a b)x]
4(2a b)
+
sin bx
2b
sin[(2a + b)x]
4(2a + b)
(82)
Z
sin2 x cos x dx =
1
3
sin3 x
(83)
Z
cos2 ax sin bx dx =
cos[(2a b)x]
4(2a b)
cos bx
2b
cos[(2a + b)x]
4(2a + b)
(84)
Z
cos2 ax sin ax dx =
1
3a
cos3 ax
(Z85)
sin2 ax cos2 bxdx =
x
4
sin 2ax
8a
sin[2(a b)x]
16(a b)
+
sin 2bx
8b
sin[2(a + b)x]
16(a + b)
(86)
Z
sin2 ax cos2 ax dx =
x
8
sin 4ax
32a
(87)
Z
tan ax dx =
1
a
ln cos ax
9

64. (88)
Z
tan2 ax dx = x +
1
a
tan ax
(89)
Z
tann ax dx =
tann+1 ax
a(1 + n)
2F1
n + 1
2
; 1;
n + 3
2
;tan2 ax
(90)
Z
tan3 axdx =
1
a
ln cos ax +
1
2a
sec2 ax
(91)
Z
sec x dx = ln j sec x + tan xj = 2 tanh1
tan
x
2
(92)
Z
sec2 ax dx =
1
a
tan ax
(93)
Z
sec3 x dx =
1
2
sec x tan x +
1
2
ln j sec x + tan xj
(94)
Z
sec x tan x dx = sec x
(95)
Z
sec2 x tan x dx =
1
2
sec2 x
(96)
Z
secn x tan x dx =
1
n
secn x; n6= 0
(97)
Z
csc x dx = ln

67. tan
x
2

70. = ln j csc x cot xj + C
10

71. (98)
Z
csc2 ax dx =
1
a
cot ax
(99)
Z
csc3 x dx =
1
2
cot x csc x +
1
2
ln j csc x cot xj
(100)
Z
cscn x cot x dx =
1
n
cscn x; n6= 0
(101)
Z
sec x csc x dx = ln j tan xj
Products of Trigonometric Functions and Mono-
mials
(102)
Z
x cos x dx = cos x + x sin x
(103)
Z
x cos ax dx =
1
a2 cos ax +
x
a
sin ax
(104)
Z
x2 cos x dx = 2x cos x +
x2 2
sin x
(105)
Z
x2 cos ax dx =
2x cos ax
a2 +
a2x2 2
a3 sin ax
(106)
Z
xn cos xdx =
1
2
(i)n+1 [(n + 1;ix) + (1)n(n + 1; ix)]
11

72. (107)
Z
xn cos ax dx =
1
2
(ia)1n [(1)n(n + 1;iax) (n + 1; ixa)]
(108)
Z
x sin x dx = x cos x + sin x
(109)
Z
x sin ax dx =
x cos ax
a
+
sin ax
a2
(110)
Z
x2 sin x dx =
2 x2
cos x + 2x sin x
(111)
Z
x2 sin ax dx =
2 a2x2
a3 cos ax +
2x sin ax
a2
(112)
Z
xn sin x dx =
1
2
(i)n [(n + 1;ix) (1)n(n + 1;ix)]
(113)
Z
x cos2 x dx =
x2
4
+
1
8
cos 2x +
1
4
x sin 2x
(114)
Z
x sin2 x dx =
x2
4
1
8
cos 2x
1
4
x sin 2x
(115)
Z
x tan2 x dx =
x2
2
+ ln cos x + x tan x
(116)
Z
x sec2 x dx = ln cos x + x tan x
12

73. Products of Trigonometric Functions and Ex-
ponentials
(117)
Z
ex sin x dx =
1
2
ex(sin x cos x)
(118)
Z
ebx sin ax dx =
1
a2 + b2 ebx(b sin ax a cos ax)
(119)
Z
ex cos x dx =
1
2
ex(sin x + cos x)
(120)
Z
ebx cos ax dx =
1
a2 + b2 ebx(a sin ax + b cos ax)
(121)
Z
xex sin x dx =
1
2
ex(cos x x cos x + x sin x)
(122)
Z
xex cos x dx =
1
2
ex(x cos x sin x + x sin x)
Integrals of Hyperbolic Functions
(123)
Z
cosh ax dx =
1
a
sinh ax
(124)
Z
eax cosh bx dx =
8
:
eax
[a cosh bx b sinh bx] a6= b
a2 b2 e2ax
x
+
a = b
4a
2
(125)
Z
sinh ax dx =
1
a
cosh ax
13

74. (126)
Z
eax sinh bx dx =
8
:
eax
[b cosh bx + a sinh bx] a6= b
a2 b2 e2ax
x
a = b
4a
2
(127)
Z
tanh axdx =
1
a
ln cosh ax
(128)
Z
eax tanh bx dx =
8
:
e(a+2b)x
(a + 2b) 2F1
h
1 +
a
2b
; 1; 2 +
a
2b
;e2bx
i
1
a
eax
2F1
h
1;
a
2b
; 1 +
a
2b
;e2bx
i
a6= b
eax 2 tan1[eax]
a
a = b
(129)
Z
cos ax cosh bx dx =
1
a2 + b2 [a sin ax cosh bx + b cos ax sinh bx]
(130)
Z
cos ax sinh bx dx =
1
a2 + b2 [b cos ax cosh bx + a sin ax sinh bx]
(131)
Z
sin ax cosh bx dx =
1
a2 + b2 [a cos ax cosh bx + b sin ax sinh bx]
(132)
Z
sin ax sinh bx dx =
1
a2 + b2 [b cosh bx sin ax a cos ax sinh bx]
(133)
Z
sinh ax cosh axdx =
1
4a
[2ax + sinh 2ax]
(134)
Z
sinh ax cosh bx dx =
1
b2 a2 [b cosh bx sinh ax a cosh ax sinh bx]
c 2014. From http://integral-table.com, last revised June 14, 2014. This mate-
rial is provided as is without warranty or representation about the accuracy, correctness or
suitability of this material for any purpose. This work is licensed under the Creative Com-
mons Attribution-Noncommercial-Share Alike 3.0 United States License. To view a copy
of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send
a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California,
94105, USA.
14