Table of Integrals∗
Basic Forms
Z
Integrals with Logarithms
1
x dx =
xn+1
n+1
Z
1
dx = ln |x|
x
Z
Z
udv = uv
vdu
Z
n
(1)
Z
Z p
(2)
(3)
1
1
dx = ln |ax + b|
ax + b
a
(4)
Z p
Integrals of Rational Functions
Z
Z
Z
(x + a)n dx =
x(x + a)n dx =
Z
Z
Z
Z
Z
1
x+a
1
(x + a)n+1 ((n + 1)x
(n + 1)(n + 2)
1
dx = tan
1 + x2
1
1
1
dx = tan
a2 + x2
a
a)
2
x
dx = x
a2 + x2
x
a
3
x
1
dx = x2
a2 + x2
2
1
a tan
1
tan
x
a
Z p
Z
Z
Z
p
x x
Z
x
a
Z r
∗
x
p
p
Z
2ax + b
p
4ac b2
2
a(x
3
a)3/2 +
✓
2b
2x
+
3a
3
Z
Z
Z
p
a)5/2
ax + b
(21)
(23)
x
dx =
a+x
p
x(a
x)
x(a + x)
a tan
a ln
⇥p
1
Z
(20)
p
x
2
dx = (x ⌥ 2a) x ± a
3
x±a
p
1
p
x+
x(a x)
x a
p
x+a
⇤
1 2
x2 ± a2 dx =
x ± a2
3
p
3/2
p
p
p
1
a2
x2
dx = sin
1
x
a
p
x
dx = x2 ± a2
2
±a
x2
x
a2
x2
dx =
p
a2
x2
(31)
(32)
(24)
(25)
Z
x
Z
1p 2
x
dx =
ax + bx + c
a
ax2 + bx + c
p
b
ln 2ax + b + 2 a(ax2 + bx + c)
2a3/2
Z
dx
x
= p
(a2 + x2 )3/2
a2 a2 + x2
◆
x, a 6= 0
ln(ax + b)
Z
Z
a2 ) dx = x ln(x2
a2 ) + a ln
(44)
x
a
2x (45)
x+a
x a
2x (46)
1
2ax + b
p
4ac b2
1
(47)
bx
1 2
x
2a
4
✓
◆
1
b2
+
x2
ln(ax + b)
2
a2
x ln(ax + b)dx =
1 2
b2 x2 dx =
x +
2
✓
◆
1
a2
x2
ln a2
2
b2
x ln a2
(33)
b2 x2
(48)
(49)
Integrals with Exponentials
Z
(34)
Z
(35)
p
(37)
1 ax
e
a
(50)
p
p
1 p ax
i ⇡
xe + 3/2 erf i ax ,
a
2a
Z x
2
2
where erf(x) = p
e t dt
⇡ 0
Z
Z
eax dx =
xeax dx =
Z
Z
Z
Z
xex dx = (x
xeax dx =
✓
x2 eax dx =
✓
x2
a
x3 ex dx = x3
xn eax
a
1)ex
x
a
x2 ex dx = x2
xn eax dx =
Z
1
a2
◆
(51)
(52)
eax
(53)
2x + 2 ex
(54)
◆
eax
(55)
6 ex
(56)
1 ax
(57)
2x
2
+ 3
a2
a
3x2 + 6x
n
a
Z
xn
e
dx
( 1)n
[1 + n, ax],
an+1
Z 1
where (a, x) =
ta 1 e t dt
xn eax dx =
(58)
x
p
p
2
i ⇡
p erf ix a
eax dx =
2 a
p
Z
p
2
⇡
e ax dx = p erf x a
2 a
Z
2
2
1
xe ax dx =
e ax
2a
Z
(40)
(41)
b
x+
a
1p
ln ax2 + bx + c dx =
4ac b2 tan
a
✓
◆
b
2x +
+ x ln ax2 + bx + c
2a
1 ⇣ p p 2
ax2 + bx + c =
2 a ax + bx + c
48a5/2
p
✓
ln(x2
p
1
1
p
dx = p ln 2ax + b + 2 a(ax2 + bx + c)
2
a
ax + bx + c
(39)
Z
(43)
Z
3b2 + 2abx + 8a(c + ax2 )
⌘
p p
+3(b3 4abc) ln b + 2ax + 2 a ax2 + bx + c
(38)
⇥
ln ax
1
dx = (ln ax)2
x
2
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan
p
x2
1 p
1
p
dx = x x2 ± a2 ⌥ a2 ln x + x2 ± a2
2
2
2
2
x ±a
(36)
p
(42)
Z
x
a2 x2
(30)
b + 2ax p 2
ax2 + bx + cdx =
ax + bx + c
4a
2
p
4ac b
+
ln 2ax + b + 2 a(ax2 + bx+ c)
8a3/2
(19)
(22)
dx =
1 2
a tan
2
x
ln(ax + b)dx =
Z p
2
(ax + b)5/2
5a
(ax + b)3/2 dx =
x2 +
Z
(16)
(18)
x
◆
1 p 2
x a
2
p
1
p
dx = ln x + x2 ± a2
x2 ± a2
(13)
(17)
2
(x
5
x
p
Z
(15)
1
dx = 2 x ± a
x±a
2 a
x2 dx =
Z
(14)
a)3/2
p
a2
(12)
p
ax + bdx =
p
2
(x
3
1
dx =
a x
adx =
Z
Z r
p
adx =
±
b3
8a5/2
b2
x p 3
+
x (ax + b)
8a2 x
3
p
p
ln a x + a(ax + b)
(28)
p
1 p
1
= x x2 ± a2 ± a2 ln x + x2 ± a2
2
2
(29)
a2 dx
Z
Integrals with Roots
x
x2
(11)
x
1
dx =
ln |ax2 + bx + c|
ax2 + bx + c
2a
b
2ax + b
p
tan 1 p
a 4ac b2
4ac b2
p
b
x3 (ax + b)dx =
12a
(10)
1
1
a+x
dx =
ln
, a 6= b
(x + a)(x + b)
b a
b+x
Z
x
a
dx =
+ ln |a + x|
(x + a)2
a+x
Z

ln axdx = x ln ax
Z
p
1
x(ax + b)dx = 3/2 (2ax + b) ax(ax + b)
4a
i
p
p
b2 ln a x + a(ax + b)
(27)
(9)
1 2
a ln |a2 + x2 |
2
b2
(7)
(8)
x
1
dx = ln |a2 + x2 |
a2 + x2
2
Z
Z p
Z
h
+
(6)
x
1
p
2
( 2b2 + abx + 3a2 x2 ) ax + b (26)
15a2
(5)
(x + a)n+1
, n 6=
n+1
1
2
dx = p
ax2 + bx + c
4ac
Z
Z
1
dx =
(x + a)2
p
x ax + bdx =
Z
2
x e
ax2
1
dx =
4
r
p
⇡
erf(x a)
a3
x
e
2a
(59)
(60)
(61)
ax2
(62)
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Integrals with Trigonometric Functions
Z
Z
Z
sin2 axdx =
1
cos ax
a
Z
3 cos ax
cos 3ax
+
4a
12a
Z
(66)
1
sin ax
a
(68)
1
cos1+p ax⇥
a(1 + p)

1+p 1 3+p
F
, ,
, cos2 ax
2 1
2
2
2
cos[(a
2(a
b)x]
b)
Z
csc xdx =
Z
cos[(a + b)x]
, a 6= b
2(a + b)
(71)
Z
Z
cos[(2a b)x]
cos ax sin bxdx =
4(2a b)
cos[(2a + b)x]
4(2a + b)
Z
Z
Z
Z
x+
Z
Z
(75)
sin 4ax
32a
Z
Z
Z
1
tan ax
a
1
⇣
Z
(79)
Z
cot x|
1
cscn x, n 6= 0
n
1
x
cos ax + sin ax
a2
a
x2 cos xdx = 2x cos x + x2
2 sin x
1 n+1
(i)
[ (n + 1, ix)
2
+( 1)n (n + 1, ix)]
tan
Z
x sin xdx =
x2 sin xdx = 2
x2 sin axdx =
x⌘
2
(82)
(83)
Z
(108)
sin x + x sin x)
(109)
Z
xex cos xdx =
1 x
e (x cos x
2
Z
2
(94)
(95)
(96)
(97)
x cos x + sin x
(99)
x cos ax
sin ax
+
a
a2
(100)
x2 cos x + 2x sin x
(101)
2 2
a x
2x sin ax
cos ax +
a3
a2
1 n
(i) [ (n + 1, ix)
2
ex sin xdx =
ebx sin axdx =
1 x
e (sin x
2
(102)
( 1)n (n + 1, ix)]
1
ebx (b sin ax
a2 + b2
cos x)
a cos ax)
Z
(93)
Products of Trigonometric Functions and
Exponentials
Z
x cos x + x sin x)
1 x
e (cos x
2
Z
(105)
1
sinh ax
a
eax sinh bxdx =
8 ax
e
>
< 2
[ b cosh bx + a sinh bx]
a
b2
2ax
x
>
:e
4a
2
eax tanh bxdx =
8 (a+2b)x
h
i
>
e
a
a
>
>
, 1, 2 + , e2bx
2 F1 1 +
>
>
2b
2b
< (a + 2b)
ha
i
1 ax
e
F
,
1,
1E,
e2bx
2
1
>
a 1 ax 2b
>
> eax 2 tan
>
[e ]
>
:
a
Z
1
tanh ax dx = ln cosh ax
a
Z
Z
Z
Z
Z
(104)
cosh axdx =
(110)
eax cosh bxdx =
8 ax
e
>
< 2
[a cosh bx b sinh bx] a 6= b
a
b2
2ax
x
>
:e
+
a=b
4a
2
Z
1
sinh axdx = cosh ax
a
(92)
(103)
(81)
(107)
xex sin xdx =
Z
(91)
1
(ia)1 n [( 1)n (n + 1, iax)
2
(n + 1, ixa)]
(98)
x sin axdx =
xn sin xdx =
(106)
1
ebx (a sin ax + b cos ax)
a2 + b2
Z
(90)
xn cosxdx =
(80)
1
1
ln cos ax +
sec2 ax
a
2a
sec2 axdx =
(88)
ebx cos axdx =
1 x
e (sin x + cos x)
2
(89)
x cos xdx = cos x + x sin x
xn cosaxdx =
Z
(78)
1
tan ax
a
sec xdx = ln | sec x + tan x| = 2 tanh
1
cot ax
a
sec x csc xdx = ln | tan x|
x cos axdx =
Z
(77)
tann+1 ax
tann axdx =
⇥
a(1 + n)
✓
◆
n+1
n+3
, 1,
, tan2 ax
2 F1
2
2
tan3 axdx =
cot x| + C
(87)
(74)
1
ln cos ax
a
tan axdx =
tan2 axdx =
x
8
(86)
2x cos ax
a2 x2 2
x cos axdx =
+
sin ax
2
a
a3
sin[2(a b)x]
x
sin 2ax
4
8a
16(a b)
sin[2(a + b)x]
sin 2bx
+
(76)
8b
16(a + b)
sin2 ax cos2 axdx =
csc axdx =
csc x cot xdx =
sin2 ax cos2 bxdx =
Z
1
sec2 x
2
2
cos bx
2b
1
cos3 ax
3a
cos2 ax sin axdx =
Z
(73)
2
Z
(72)
1
sin3 x
3
Z
ex cos xdx =
Integrals of Hyperbolic Functions
2
n
Z
(84)
(85)
1
1
cot x csc x + ln | csc x
2
2
3
2
sin2 x cos xdx =
sec2 x tan xdx =
Z
Products of Trigonometric Functions and
Monomials
(70)
sin[(2a b)x]
sin ax cos bxdx =
4(2a b)
sin[(2a + b)x]
sin bx
+
2b
4(2a + b)
sec x tan xdx = sec x
1
sec x tan xdx = secn x, n 6= 0
n
(69)
3 sin ax
sin 3ax
+
4a
12a
Z
n
(67)
Z
1
1
sec x tan x + ln | sec x + tan x|
2
2
x
csc xdx = ln tan
= ln | csc x
2
cosp axdx =
cos3 axdx =
Z
Z
(65)
x
sin 2ax
cos2 axdx = +
2
4a
Z
Z
(64)
1 1 n 3
,
, , cos2 ax
2
2
2
cos axdx =
cos ax sin bxdx =
Z
2 F1

sin3 axdx =
Z
Z
sin 2ax
4a
sin axdx =
Z
Z
x
2
(63)
sec3 x dx =
n
Z
Z
1
cos ax
a
sin axdx =
Z
Z
a 6= b
(111)
(112)
(113)
a=b
a 6= b (114)
a=b
(115)
1
[a sin ax cosh bx
a2 + b2
+b cos ax sinh bx]
(116)
cos ax cosh bxdx =
1
[b cos ax cosh bx+
a2 + b2
a sin ax sinh bx]
(117)
cos ax sinh bxdx =
1
[ a cos ax cosh bx+
a2 + b2
b sin ax sinh bx]
(118)
sin ax cosh bxdx =
1
[b cosh bx sin ax
a2 + b2
a cos ax sinh bx]
sin ax sinh bxdx =
sinh ax cosh axdx =
sinh ax cosh bxdx =
1
[ 2ax + sinh 2ax]
4a
(119)
(120)
1
[b cosh bx sinh ax
b2 a2
a cosh ax sinh bx]
(121)