Table of Basic Integrals Basic Forms

Table of Basic Integrals

Basic Forms

(1)

xndx = 1 xn+1, n = -1

n+1

1

(2)

dx = ln |x|

x

(3)

udv = uv - vdu

1

1

(4)

dx = ln |ax + b|

ax + b a

Integrals of Rational Functions

1

1

(5)

dx = -

(x + a)2

x+a

(6)

(x

+ a)ndx

=

(x

+

a)n+1 ,n

=

-1

n+1

(7)

x(x + a)ndx = (x + a)n+1((n + 1)x - a)

(n + 1)(n + 2)

(8)

1

1 +

x2

dx

=

tan-1

x

(9)

1 dx = 1 tan-1 x

a2 + x2

a

a

1

(10)

x dx = 1 ln |a2 + x2|

a2 + x2

2

(11)

a2

x2 +

x2

dx

=

x

-

a

tan-1

x a

(12)

x3 dx = 1 x2 - 1 a2 ln |a2 + x2|

a2 + x2

22

(13)

ax2

1 + bx +

dx c

=

2

4ac - b2

tan-1

2ax + b

4ac - b2

1

1 a+x

(14)

dx =

ln

, a=b

(x + a)(x + b) b - a b + x

x

a

(15)

(x + a)2 dx = a + x + ln |a + x|

(16)

x

dx = 1 ln |ax2+bx+c|- b

tan-1

2ax

+

b

ax2 + bx + c

2a

a 4ac - b2

4ac - b2

Integrals with Roots

(17)

x

-

a

dx

=

2 (x

-

a)3/2

3

1

(18)

dx = 2 x ? a

x?a

(19)

1

dx = -2 a - x

a-x

2

2a 3

(x

-

a)3/2

+

2 5

(x

-

a)5/2

,

or

(20)

x x - a dx =

2 3

x(x

-

a)3/2

-

4 15

(x

-

a)5/2

,

or

2 15

(2a

+

3x)(x

-

a)3/2

2b 2x

(21)

ax + b dx = +

ax + b

3a 3

(22)

(ax + b)3/2 dx = 2 (ax + b)5/2

5a

x

2

(23)

dx = (x 2a) x ? a

x?a

3

(24)

x dx = - x(a - x) - a tan-1 x(a - x)

a-x

x-a

x

(25)

dx = x(a + x) - a ln x + x + a

a+x

(26)

x ax

+

b

dx

=

2 15a2

(-2b2

+

abx

+

3a2x2) ax

+

b

(27)

1 x(ax + b) dx =

(2ax + b)

ax(ax + b) - b2 ln

a x+

a(ax + b)

4a3/2

(28)

b

b2 x

x3(ax + b) dx = 12a - 8a2x + 3

b3

x3(ax + b)+ 8a5/2 ln a x + a(ax + b)

(29)

x2 ? a2

dx =

1

x x2

?

a2

?

1 a2 ln

x + x2 ? a2

2

2

3

(30)

a2

-

x2

dx

=

1

x a2

-

x2

+

1 a2

tan-1

x

2

2

a2 - x2

(31)

1

x x2 ? a2 dx =

x2 ? a2 3/2

3

(32)

1

dx = ln x + x2 ? a2

x2 ? a2

(33)

1

dx = sin-1 x

a2 - x2

a

x

(34)

dx = x2 ? a2

x2 ? a2

(35)

x

dx = - a2 - x2

a2 - x2

(36)

x2

dx =

1

x x2

?

a2

1 a2 ln

x + x2 ? a2

x2 ? a2

2

2

(37)

ax2 + bx + c dx =

b

+

2ax

ax2

+

bx

+

4ac c+

-

b2

ln

2ax + b + 2

a(ax2 + bx+c)

4a

8a3/2

(38)

1

x ax2 + bx + c dx = 48a5/2

2 a ax2 + bx + c

-3b2 + 2abx + 8a(c + ax2)

+3(b3 - 4abc) ln

b + 2ax + 2 a ax2 + bx + c

4

(39)

1

1 dx = ln 2ax + b + 2

a(ax2 + bx + c)

ax2 + bx + c

a

(40)

x

1

b

dx =

ax2 + bx + c

a

ax2 + bx + c- 2a3/2 ln 2ax + b + 2

a(ax2 + bx + c)

dx

x

(41)

(a2

+ x2)3/2

=

a2 a2

+ x2

Integrals with Logarithms

(42)

ln ax dx = x ln ax - x

(43)

x ln x dx = 1 x2 ln x - x2

2

4

(44)

x2 ln x dx = 1 x3 ln x - x3

3

9

(45)

xn ln x dx = xn+1

ln x

1

n + 1 - (n + 1)2

,

n = -1

(46)

ln ax dx = 1 (ln ax)2

x

2

ln x

1 ln x

(47)

x2

dx = - - x

x

5

b

(48)

ln(ax + b) dx = x + ln(ax + b) - x, a = 0

a

(49)

ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan-1 x - 2x

a

(50)

ln(x2 - a2) dx = x ln(x2 - a2) + a ln x + a - 2x

x-a

(51)

ln ax2 + bx + c

dx

=

1 4ac

-

b2 tan-1

2ax

+b

-2x+

b +x

ln ax2 + bx + c

a

4ac - b2

2a

(52)

x ln(ax + b) dx = bx - 1 x2 + 1 x2 - b2 ln(ax + b)

2a 4 2

a2

(53)

x ln a2 - b2x2 dx = - 1 x2 + 1 x2 - a2 ln a2 - b2x2

22

b2

(54)

(ln x)2 dx = 2x - 2x ln x + x(ln x)2

(55)

(ln x)3 dx = -6x + x(ln x)3 - 3x(ln x)2 + 6x ln x

(56)

x(ln x)2 dx = x2 + 1 x2(ln x)2 - 1 x2 ln x

42

2

(57)

x2(ln x)2 dx = 2x3 + 1 x3(ln x)2 - 2 x3 ln x

27 3

9

6

Integrals with Exponentials

(58)

eax dx = 1 eax

a

(59) xeax dx = 1 xeax +

i

erf

i ax

,

where erf(x) = 2

x

e-t2 dt

a

2a3/2

0

(60)

xex dx = (x - 1)ex

(61)

xeax dx =

x1 -

eax

a a2

(62)

x2ex dx = x2 - 2x + 2 ex

(63)

x2eax dx =

x2 2x 2 a - a2 + a3

eax

(64)

x3ex dx = x3 - 3x2 + 6x - 6 ex

(65)

xneax dx = xneax - n aa

xn-1eax dx

(66)

xneax

dx

=

(-1)n an+1 [1 + n, -ax],

where

(a, x) =

ta-1e-t dt

x

(67)

eax2 dx = - i erf

ix a

2a

7

(68)

e-ax2 dx =

erf x a

2a

(69)

xe-ax2 dx = - 1 e-ax2 2a

(70)

x2e-ax2 dx = 1

erf(x a) -

x e-ax2

4 a3

2a

Integrals with Trigonometric Functions

1

(71)

sin ax dx = - cos ax

a

(72)

sin2 ax dx = x - sin 2ax

2 4a

(73)

sin3

ax

dx

=

3 -

cos

ax

+

cos

3ax

4a

12a

(74)

sinn ax

dx

1 = - cos ax

a

2F1

1 , 1 - n , 3 , cos2 ax 222

1

(75)

cos ax dx = sin ax

a

(76)

cos2 ax dx = x + sin 2ax

2 4a

(77)

cos3 axdx = 3 sin ax + sin 3ax

4a

12a

8

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