Table of Basic Integrals
Basic Forms
Z
(1)
xn dx =
Z
(2)
1
xn+1 , n 6= −1
n+1
1
dx = ln |x|
x
Z
Z
udv = uv −
(3)
Z
vdu
1
1
dx = ln |ax + b|
ax + b
a
(4)
Integrals of Rational Functions
Z
1
1
dx = −
2
(x + a)
x+a
(5)
Z
(6)
Z
(7)
(x + a)n+1
, n 6= −1
(x + a) dx =
n+1
n
x(x + a)n dx =
Z
(8)
Z
(9)
(x + a)n+1 ((n + 1)x − a)
(n + 1)(n + 2)
1
dx = tan−1 x
1 + x2
1
1
−1 x
dx
=
tan
a2 + x 2
a
a
1
Z
(10)
a2
x2
x
dx = x − a tan−1
2
2
a +x
a
Z
(11)
x3
1 2 1 2
x − a ln |a2 + x2 |
dx
=
a2 + x 2
2
2
Z
(12)
Z
(13)
1
2
−1 2ax + b
√
√
tan
dx
=
ax2 + bx + c
4ac − b2
4ac − b2
Z
(14)
1
a+x
1
dx =
ln
, a 6= b
(x + a)(x + b)
b−a b+x
Z
(15)
Z
(16)
ax2
x
1
dx = ln |a2 + x2 |
2
+x
2
a
x
dx =
+ ln |a + x|
2
(x + a)
a+x
x
1
b
2ax + b
dx =
ln |ax2 +bx+c|− √
tan−1 √
+ bx + c
2a
a 4ac − b2
4ac − b2
Integrals with Roots
Z
(17)
Z
(18)
Z
(19)
√
2
x − a dx = (x − a)3/2
3
√
√
√
1
dx = 2 x ± a
x±a
√
1
dx = −2 a − x
a−x
2
√
Z
x x − a dx =
(20)



Z √
(21)
Z
(22)
Z
(23)
Z r
(24)
Z
(26)
(27)
Z p
(28)
Z p
(29)
ax + b dx =
2b 2x
+
3a
3
(ax + b)3/2 dx =
or
or
√
ax + b
2
(ax + b)5/2
5a
√
x
2
√
dx = (x ∓ 2a) x ± a
3
x±a
p
x
dx = − x(a − x) − a tan−1
a−x
Z r
(25)
2a
(x − a)3/2 + 25 (x − a)5/2 ,
3
4
2
x(x − a)3/2 − 15
(x − a)5/2 ,
3
2
(2a + 3x)(x − a)3/2
15
p
x(a − x)
x−a
p
√
√
x
dx = x(a + x) − a ln x + x + a
a+x
√
x ax + b dx =
x(ax + b) dx =
√
2
2
2 2
(−2b
+
abx
+
3a
x
)
ax + b
15a2
i
p
p
√
1 h
2
(2ax
+
b)
ax(ax
+
b)
−
b
ln
a
x
+
a(ax
+
b)
4a3/2
2
p
p
√
b
b
x
b3
3
3
x (ax + b) dx =
− 2 +
x (ax + b)+ 5/2 ln a x + a(ax + b)
12a 8a x 3
8a
Z √
√
1 √
1
x2 ± a2 dx = x x2 ± a2 ± a2 ln x + x2 ± a2
2
2
3
Z √
1 √
1
x
a2 − x2 dx = x a2 − x2 + a2 tan−1 √
2
2
a2 − x 2
(30)
√
3/2
1 2
x ± a2
x x2 ± a2 dx =
3
Z
(31)
Z
(32)
√
√
1
dx = ln x + x2 ± a2
x 2 ± a2
Z
(33)
Z
(34)
Z
(35)
Z
(36)
(37)
Z √
ax2
√
√
√
√
a2
1
x
dx = sin−1
2
a
−x
√
x
dx = x2 ± a2
x 2 ± a2
√
x
dx = − a2 − x2
a2 − x 2
√
1 √
x2
1
dx = x x2 ± a2 ∓ a2 ln x + x2 ± a2
2
2
x 2 ± a2
p
b + 2ax √ 2
4ac − b2
ln
2ax
+
b
+
2
+ bx + c dx =
ax + bx + c+
a(ax2 + bx+ c)
4a
8a3/2
(38)
Z √
x ax2 + bx + c dx =
1 √ √ 2
2
2
2
a
ax
+
bx
+
c
−3b
+
2abx
+
8a(c
+
ax
)
48a5/2
√ √
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + c
4
Z
(39)
(40)
Z
√
√
p
1
1
dx = √ ln 2ax + b + 2 a(ax2 + bx + c)
a
ax2 + bx + c
p
x
1√ 2
b
dx =
ax + bx + c− 3/2 ln 2ax + b + 2 a(ax2 + bx + c)
a
2a
ax2 + bx + c
Z
(41)
dx
x
√
=
(a2 + x2 )3/2
a2 a2 + x 2
Integrals with Logarithms
Z
ln ax dx = x ln ax − x
(42)
Z
x2
1
x ln x dx = x2 ln x −
2
4
Z
1 3
x3
x ln x dx = x ln x −
3
9
(43)
(44)
Z
(45)
2
n
x ln x dx = x
ln x
1
−
n + 1 (n + 1)2
Z
ln ax
1
dx = (ln ax)2
x
2
Z
ln x
1 ln x
dx = − −
2
x
x
x
(46)
(47)
n+1
5
,
n 6= −1
Z
(48)
Z
(49)
Z
(50)
b
ln(ax + b) dx = x +
ln(ax + b) − x, a 6= 0
a
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
x
− 2x
a
x+a
− 2x
x−a
(51)
Z
1√
b
2
−1 2ax + b
2
ln ax + bx + c dx =
4ac − b tan √
−2x+
+ x ln ax2 + bx + c
a
2a
4ac − b2
Z
(52)
bx 1 2 1
x ln(ax + b) dx =
− x +
2a 4
2
Z
(53)
2
x ln a − b x
Z
(54)
Z
(55)
1
1
dx = − x2 +
2
2
a2
2
x − 2 ln a2 − b2 x2
b
(ln x)2 dx = 2x − 2x ln x + x(ln x)2
(ln x)3 dx = −6x + x(ln x)3 − 3x(ln x)2 + 6x ln x
Z
(56)
Z
(57)
2 2
b2
2
x − 2 ln(ax + b)
a
x(ln x)2 dx =
x2 1 2
1
+ x (ln x)2 − x2 ln x
4
2
2
x2 (ln x)2 dx =
2x3 1 3
2
+ x (ln x)2 − x3 ln x
27
3
9
6
Integrals with Exponentials
Z
(58)
(59)
Z
√
ax
xe
√
Z x
√ 1 √ ax
i π
2
2
e−t dt
dx =
xe + 3/2 erf i ax , where erf(x) = √
a
2a
π 0
Z
xex dx = (x − 1)ex
(60)
Z
ax
(61)
xe
Z
Z
(63)
(64)
Z
(65)
(66)
2 ax
xe
Z
x e
n ax
eax
dx =
2
x2 2x
− 2 + 3
a
a
a
eax
xn eax n
dx =
−
a
a
Z
xn−1 eax dx
(−1)n
dx = n+1 Γ[1 + n, −ax], where Γ(a, x) =
a
Z
(67)
dx =
x
1
− 2
a a
x3 ex dx = x3 − 3x2 + 6x − 6 ex
x e
n ax
x2 ex dx = x2 − 2x + 2 ex
(62)
Z
1
eax dx = eax
a
e
ax2
√
√ i π
dx = − √ erf ix a
2 a
7
Z
x
∞
ta−1 e−t dt
Z
−ax2
(68)
e
Z
2
xe−ax dx = −
(69)
Z
(70)
√
√ π
dx = √ erf x a
2 a
1
dx =
4
2 −ax2
xe
r
1 −ax2
e
2a
√
x
π
2
a) − e−ax
erf(x
3
a
2a
Integrals with Trigonometric Functions
Z
(71)
Z
(72)
Z
(74)
sin2 ax dx =
sin3 ax dx = −
(73)
Z
1
sin ax dx = − cos ax
a
x sin 2ax
−
2
4a
3 cos ax cos 3ax
+
4a
12a
1
1 1−n 3
2
sin ax dx = − cos ax 2 F1 ,
, , cos ax
a
2
2
2
n
Z
(75)
cos ax dx =
Z
(76)
Z
(77)
cos2 ax dx =
cos3 axdx =
1
sin ax
a
x sin 2ax
+
2
4a
3 sin ax sin 3ax
+
4a
12a
8
Z
(78)
1+p 1 3+p
1
1+p
2
cos ax × 2 F1
, ,
, cos ax
cos axdx = −
a(1 + p)
2 2 2
p
Z
(79)
cos x sin x dx =
1 2
1
1
sin x + c1 = − cos2 x + c2 = − cos 2x + c3
2
2
4
Z
(80)
cos ax sin bx dx =
Z
(81)
sin2 ax cos bx dx = −
Z
(82)
Z
(83)
cos[(a − b)x] cos[(a + b)x]
−
, a 6= b
2(a − b)
2(a + b)
sin2 x cos x dx =
cos2 ax sin bx dx =
Z
(84)
sin[(2a − b)x] sin bx sin[(2a + b)x]
+
−
4(2a − b)
2b
4(2a + b)
1 3
sin x
3
cos[(2a − b)x] cos bx cos[(2a + b)x]
−
−
4(2a − b)
2b
4(2a + b)
cos2 ax sin ax dx = −
1
cos3 ax
3a
(85)
Z
x sin 2ax sin[2(a − b)x] sin 2bx sin[2(a + b)x]
sin2 ax cos2 bxdx = −
−
+
−
4
8a
16(a − b)
8b
16(a + b)
Z
(86)
sin2 ax cos2 ax dx =
Z
(87)
x sin 4ax
−
8
32a
1
tan ax dx = − ln cos ax
a
9
Z
tan2 ax dx = −x +
(88)
Z
(89)
tann+1 ax
tan ax dx =
× 2 F1
a(1 + n)
n
Z
(90)
Z
tan3 axdx =
1
tan ax
a
n+1
n+3
, 1,
, − tan2 ax
2
2
1
1
ln cos ax +
sec2 ax
a
2a
x
sec x dx = ln | sec x + tan x| = 2 tanh−1 tan
2
(91)
Z
(92)
Z
(93)
sec3 x dx =
sec2 ax dx =
1
tan ax
a
1
1
sec x tan x + ln | sec x + tan x|
2
2
Z
(94)
sec x tan x dx = sec x
Z
(95)
Z
(96)
sec2 x tan x dx =
secn x tan x dx =
Z
(97)
csc x dx = ln tan
1
sec2 x
2
1
secn x, n 6= 0
n
x
= ln | csc x − cot x| + C
2
10
Z
(98)
Z
(99)
1
csc2 ax dx = − cot ax
a
1
1
csc3 x dx = − cot x csc x + ln | csc x − cot x|
2
2
Z
(100)
1
cscn x cot x dx = − cscn x, n 6= 0
n
Z
sec x csc x dx = ln | tan x|
(101)
Products of Trigonometric Functions and Monomials
Z
(102)
x cos x dx = cos x + x sin x
Z
(103)
x cos ax dx =
Z
(104)
Z
(105)
Z
(106)
1
x
cos ax + sin ax
2
a
a
x2 cos x dx = 2x cos x + x2 − 2 sin x
x2 cos ax dx =
2x cos ax a2 x2 − 2
+
sin ax
a2
a3
1
xn cos xdx = − (i)n+1 [Γ(n + 1, −ix) + (−1)n Γ(n + 1, ix)]
2
11
Z
(107)
1
xn cos ax dx = (ia)1−n [(−1)n Γ(n + 1, −iax) − Γ(n + 1, ixa)]
2
Z
x sin x dx = −x cos x + sin x
(108)
Z
x sin ax dx = −
(109)
Z
(110)
Z
Z
(112)
(113)
Z
(114)
Z
x cos2 x dx =
1
x2 1
+ cos 2x + x sin 2x
4
8
4
x sin2 x dx =
x2 1
1
− cos 2x − x sin 2x
4
8
4
x tan2 x dx = −
Z
(116)
2x sin ax
2 − a2 x 2
cos ax +
3
a
a2
1
xn sin x dx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)]
2
Z
(115)
x2 sin x dx = 2 − x2 cos x + 2x sin x
x2 sin ax dx =
(111)
x cos ax sin ax
+
a
a2
x2
+ ln cos x + x tan x
2
x sec2 x dx = ln cos x + x tan x
12
Products of Trigonometric Functions and Exponentials
Z
(117)
Z
(118)
ebx sin ax dx =
Z
(119)
Z
(120)
a2
1
ebx (b sin ax − a cos ax)
+ b2
1
ex cos x dx = ex (sin x + cos x)
2
ebx cos ax dx =
a2
1
ebx (a sin ax + b cos ax)
+ b2
Z
1
xex sin x dx = ex (cos x − x cos x + x sin x)
2
Z
1
xex cos x dx = ex (x cos x − sin x + x sin x)
2
(121)
(122)
1
ex sin x dx = ex (sin x − cos x)
2
Integrals of Hyperbolic Functions
Z
(123)
cosh ax dx =
Z
(124)
 ax
e

 2
[a cosh bx − b sinh bx] a 6= b
2
eax cosh bx dx = a2ax− b

e + x
a=b
4a
2
Z
(125)
1
sinh ax
a
sinh ax dx =
13
1
cosh ax
a
Z
(126)
(127)
 ax
e

 2
[−b cosh bx + a sinh bx] a 6= b
2
ax
e sinh bx dx = a2ax− b

e − x
a=b
4a
2
Z
1
tanh ax dx = ln cosh ax
a
 (a+2b)x
h
i
e
a
a

2bx


F
1
+
,
1,
2
+
,
−e
2
1


Z
2b
 (a + 2b)
h 2b a
i
1
a
ax
ax
2bx
(128)
e tanh bx dx =
− e 2 F1 1, , 1 + , −e
a 6= b

a −1 ax 2b
2b


ax


 e − 2 tan [e ]
a=b
a
Z
1
[a sin ax cosh bx + b cos ax sinh bx]
(129)
cos ax cosh bx dx = 2
a + b2
Z
(130)
cos ax sinh bx dx =
a2
Z
(131)
sin ax cosh bx dx =
a2
1
[−a cos ax cosh bx + b sin ax sinh bx]
+ b2
Z
(132)
sin ax sinh bx dx =
1
[b cos ax cosh bx + a sin ax sinh bx]
+ b2
a2
1
[b cosh bx sin ax − a cos ax sinh bx]
+ b2
Z
(133)
sinh ax cosh axdx =
Z
(134)
sinh ax cosh bx dx =
b2
1
[−2ax + sinh 2ax]
4a
1
[b cosh bx sinh ax − a cosh ax sinh bx]
− a2
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