Table of Integrals

Table of Integrals

Basic Forms

xndx = 1 xn+1

(1)

n+1

1 dx = ln |x|

(2)

x

udv = uv - vdu

(3)

1 dx = 1 ln |ax + b|

(4)

ax + b

a

Integrals of Rational Functions

1

1

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

(5)

(x + a)ndx

=

(x + a)n+1 ,n

=

-1

(6)

n+1

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

(7)

(n + 1)(n + 2)

x ax + bdx =

2 15a2

(-2b2

+

abx

+

3a2x2

) ax

+

b

(26)

1 x(ax + b)dx = 4a3/2 (2ax + b) ax(ax + b)

-b2 ln

a x+

a(ax + b)

(27)

x3(ax + b)dx =

b

b2

x

12a - 8a2x + 3

x3(ax + b)

b3

+ 8a5/2 ln a x + a(ax + b) (28)

x2 ? a2dx = 1 x x2 ? a2 ? 1 a2 ln x +

2

2

x2 ? a2 (29)

Integrals with Logarithms

ln axdx = x ln ax - x

(42)

ln ax dx = 1 (ln ax)2

(43)

x

2

ln(ax + b)dx = x + b ln(ax + b) - x, a = 0 (44) a

ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan-1 x - 2x (45) a

ln(x2 - a2) dx = x ln(x2 - a2) + a ln x + a - 2x (46) x-a

ln

ax2 + bx + c

1 dx =

4ac - b2 tan-1 2ax + b

a

4ac - b2

- 2x + b + x ln ax2 + bx + c

(47)

2a

1

1 + x2

dx

=

tan-1

x

(8)

a2

1 + x2 dx

=

1 a

tan-1

x a

(9)

a2

x + x2 dx

=

1 2

ln |a2

+ x2|

(10)

x2 dx = x - a tan-1 x

(11)

a2 + x2

a

x3 a2 + x2 dx

=

1 x2 2

-

1 a2 2

ln |a2

+

x2|

(12)

ax2

1

dx

+ bx + c

=

2

4ac - b2

tan-1

2ax + b

4ac - b2

(13)

1

1 a+x

dx =

ln

, a = b (14)

(x + a)(x + b)

b-a b+x

x

a

dx =

+ ln |a + x|

(15)

(x + a)2

a+x

ax2

x dx

+ bx + c

=

1 2a

ln |ax2

+ bx + c|

-b

tan-1 2ax + b (16)

a 4ac - b2

4ac - b2

Integrals with Roots

x - adx

=

2 (x - a)3/2

(17)

3

1

dx = 2 x ? a

(18)

x?a

1

dx = -2 a - x

(19)

a-x

x x-

adx

=

2 a(x

-

a)3/2

+

2 (x

- a)5/2

(20)

3

5

2b 2x

ax + bdx = +

ax + b

(21)

3a 3

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

(22)

5a

x

dx

=

2 (x

2a) x

?

a

(23)

x?a

3

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

x(a - x) (24)

a-x

x-a

a2 - x2dx = 1 x a2 - x2 + 1 a2 tan-1 x

2

2

a2 - x2

(30)

x x2 ? a2dx = 1 x2 ? a2 3/2 3

(31)

1

dx = ln x + x2 ? a2

(32)

x2 ? a2

1 dx = sin-1 x

(33)

a2 - x2

a

x dx = x2 ? a2

(34)

x2 ? a2

x dx = - a2 - x2

(35)

a2 - x2

x2

1 dx = x

x2 ? a2 1 a2 ln x +

x2 ? a2

2

2

x2 ? a2 (36)

ax2 + bx + cdx = b + 2ax ax2 + bx + c 4a

4ac - b2

+

ln 2ax + b + 2

a(ax2 + bx+c)

(37)

8a3/2

x ax2 + bx + c = 1

2 a ax2 + bx + c

48a5/2

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

+3(b3 - 4abc) ln

b + 2ax + 2 a

ax2 + bx + c

(38)

1

dx = 1 ln 2ax + b + 2 a(ax2 + bx + c)

ax2 + bx + c

a

(39)

x

1 dx =

ax2 + bx + c

ax2 + bx + c

a

b - 2a3/2 ln 2ax + b + 2

a(ax2 + bx + c)

(40)

x ln(ax + b)dx = bx - 1 x2 2a 4

+ 1 x2 - b2 ln(ax + b)

(48)

2

a2

x ln a2 - b2x2 dx = - 1 x2+ 2

1 x2 - a2

2

b2

ln a2 - b2x2

(49)

Integrals with Exponentials

eaxdx = 1 eax a

(50)

xeaxdx

=

1 xeax a

+

i 2a3/2 erf

i ax

,

where erf(x) = 2 x e-t2 dt

(51)

0

xexdx = (x - 1)ex

(52)

xeaxdx =

x1 a - a2

eax

(53)

x2exdx = x2 - 2x + 2 ex

(54)

x2eaxdx =

x2 2x 2 -+

eax

a a2 a3

x3exdx = x3 - 3x2 + 6x - 6 ex

xneax dx = xneax - n aa

xn-1eax dx

(55) (56) (57)

xneax

dx

=

(-1)n an+1 [1

+

n, -ax],

where (a, x) =

ta-1e-t dt

x

eax2 dx = - i erf

ix a

2a

e-ax2 dx =

erf

xa

2a

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

(58)

(59) (60) (61)

x

dx = x(a + x) - a ln x + x + a (25)

a+x

dx

x

=

(a2 + x2)3/2 a2 a2 + x2

(41)

x2e-ax2 dx = 1

erf(x a)

-

x e-ax2

4 a3

2a

(62)

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Integrals with Trigonometric Functions

sin axdx = - 1 cos ax

(63)

a

sin2 axdx = x - sin 2ax

(64)

2 4a

sinn axdx =

1

- cos ax a

2 F1

1 , 1 - n , 3 , cos2 ax 222

(65)

sin3

axdx

=

3 cos ax -

+

cos 3ax

(66)

4a

12a

cos axdx = 1 sin ax

(67)

a

cos2 axdx = x + sin 2ax

(68)

2 4a

cosp axdx = - 1 cos1+p ax? a(1 + p)

2 F1

1 + p , 1 , 3 + p , cos2 ax 222

(69)

cos3 axdx = 3 sin ax + sin 3ax

(70)

4a

12a

cos ax sin bxdx = cos[(a - b)x] - cos[(a + b)x] , a = b

2(a - b)

2(a + b)

(71)

sin2 ax cos bxdx = - sin[(2a - b)x] 4(2a - b)

+ sin bx - sin[(2a + b)x]

(72)

2b

4(2a + b)

sin2 x cos xdx = 1 sin3 x

(73)

3

cos2 ax sin bxdx = cos[(2a - b)x] - cos bx

4(2a - b)

2b

cos[(2a + b)x]

-

(74)

4(2a + b)

cos2 ax sin axdx = - 1 cos3 ax

(75)

3a

sin2 ax cos2 bxdx = x - sin 2ax - sin[2(a - b)x]

4 8a

16(a - b)

+ sin 2bx - sin[2(a + b)x]

(76)

8b

16(a + b)

sin2 ax cos2 axdx = x - sin 4ax

(77)

8 32a

tan axdx = - 1 ln cos ax

(78)

a

tan2 axdx = -x + 1 tan ax

(79)

a

tann axdx

=

tann+1 ax ?

a(1 + n)

2 F1

n + 1 , 1, n + 3 , - tan2 ax

2

2

(80)

tan3 axdx = 1 ln cos ax + 1 sec2 ax

(81)

a

2a

sec xdx = ln | sec x + tan x| = 2 tanh-1 tan x (82) 2

sec2 axdx = 1 tan ax

(83)

a

sec3 x dx = 1 sec x tan x + 1 ln | sec x + tan x| (84)

2

2

sec x tan xdx = sec x

(85)

sec2 x tan xdx = 1 sec2 x

(86)

2

secn x tan xdx = 1 secn x, n = 0

(87)

n

x csc xdx = ln tan = ln | csc x - cot x| + C (88)

2

csc2 axdx = - 1 cot ax

(89)

a

csc3 xdx = - 1 cot x csc x + 1 ln | csc x - cot x| (90)

2

2

cscn x cot xdx = - 1 cscn x, n = 0

(91)

n

sec x csc xdx = ln | tan x|

(92)

Products of Trigonometric Functions and Monomials

x cos xdx = cos x + x sin x

(93)

1

x

x cos axdx = a2 cos ax + a sin ax

(94)

x2 cos xdx = 2x cos x + x2 - 2 sin x

(95)

x2 cos axdx

=

2x cos ax a2

+

a2x2 - 2 a3

sin ax

(96)

xncosxdx = - 1 (i)n+1 [(n + 1, -ix) 2

+(-1)n(n + 1, ix)]

(97)

xncosaxdx = 1 (ia)1-n [(-1)n(n + 1, -iax) 2

-(n + 1, ixa)]

(98)

x sin xdx = -x cos x + sin x

x cos ax sin ax

x sin axdx = - a

+ a2

x2 sin xdx = 2 - x2 cos x + 2x sin x

(99) (100) (101)

x2 sin axdx = 2 - a2x2 cos ax + 2x sin ax

a3

a2

(102)

xn sin xdx = - 1 (i)n [(n + 1, -ix) - (-1)n(n + 1, -ix)] 2 (103)

Products of Trigonometric Functions and Exponentials

ex sin xdx = 1 ex(sin x - cos x) 2

(104)

ebx

sin axdx

=

a2

1 +

b2 ebx(b sin ax

-

a cos ax)

(105)

ex cos xdx = 1 ex(sin x + cos x) 2

(106)

ebx cos axdx = 1 ebx(a sin ax + b cos ax) (107) a2 + b2

xex sin xdx = 1 ex(cos x - x cos x + x sin x) (108) 2

xex cos xdx = 1 ex(x cos x - sin x + x sin x) (109) 2

Integrals of Hyperbolic Functions

cosh axdx = 1 sinh ax a

(110)

eax cosh bxdx =

eax

a2

-

b2

[a

cosh

bx

-

b

sinh

bx]

e2ax x

+

4a 2

a=b a=b

sinh axdx = 1 cosh ax a

(111) (112)

eax sinh bxdx =

eax

a2

-

b2

[-b

cosh

bx

+

a

sinh

bx]

e2ax x

-

4a 2

a=b a=b

(113)

eax tanh bxdx =

e(a+2b)x

(a

+

2b)

2

F1

1+

a , 1, 2 +

a , -e2bx

2b

2b

-

1 a

eax

2

F1

a , 1, 1E, -e2bx 2b

eax

-

2

tan-1[eax]

a

a = b (114) a=b

1 tanh ax dx = ln cosh ax

a

(115)

1 cos ax cosh bxdx = a2 + b2 [a sin ax cosh bx

+b cos ax sinh bx]

(116)

1 cos ax sinh bxdx = a2 + b2 [b cos ax cosh bx+

a sin ax sinh bx]

(117)

1

sin ax cosh bxdx =

[-a cos ax cosh bx+

a2 + b2

b sin ax sinh bx]

(118)

1

sin ax sinh bxdx =

[b cosh bx sin ax-

a2 + b2

a cos ax sinh bx]

(119)

1 sinh ax cosh axdx = [-2ax + sinh 2ax]

4a

(120)

1 sinh ax cosh bxdx = b2 - a2 [b cosh bx sinh ax

-a cosh ax sinh bx]

(121)

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