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Table of Integrals

Basic Forms

(1)

Z

xndx = 1

n + 1x

n+1

, n 6= −1

(2)

Z 1

xdx = ln |x|

(3)

Z udv = uv −

Z vdu

(4)

Z 1

ax + bdx =

1

aln |ax + b|

Integrals of Rational Functions

(5)

Z 1 (x + a)2dx = − 1

x + a

(6)

Z (x + a)ndx = (x + a)

n+1

n + 1 , n 6= −1

(7)

Z x(x + a)ndx = (x + a)

n+1((n + 1)x − a) (n + 1)(n + 2)

(8)

Z 1

1 + x2dx = tan−1x

(9)

a2+ x2dx = 1

atan

−1 x a

(10) x

a2+ x2dx = 1

2ln |a

2+ x2|

(11)

Z

x2

a2 + x2dx = x − a tan−1 x

a

(12)

Z

x3

a2+ x2dx = 1

2x

2− 1

2a

2ln |a2+ x2|

(13)

ax2 + bx + cdx =

2

√ 4ac − b2 tan−1 √2ax + b

4ac − b2

(14)

(x + a)(x + b)dx =

1

b − aln

a + x

b + x, a 6= b

(15)

(x + a)2dx = a

a + x + ln |a + x|

(16)

Z

x

ax2+ bx + cdx =

1 2aln |ax

2

a√ 4ac − b2 tan−1√2ax + b

4ac − b2

Integrals with Roots

(17)

x − a dx = 2

3(x − a)

3/2

(18)

Z 1

√

x ± a dx = 2

√

x ± a

(19)

√

a − x dx = −2

√

a − x

Z

x√

x − a dx =





2a

3 (x − a)3/2+25 (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)

ax + b dx = 2b

3a +

2x 3

√

ax + b

(22)

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

5a(ax + b)

5/2

(23)

Z x

√

x ± a dx =

2

3(x ∓ 2a)

√

x ± a

(24)

x

a − x dx = −

p x(a − x) − a tan−1 px(a − x)

x − a

(25)

x

a + x dx =

p x(a + x) − a ln√x +√x + a

(26)

Z

x√

ax + b dx = 2

15a2(−2b2+ abx + 3a2x2)√

ax + b

(27)

Z

p

x(ax + b) dx = 1

4a3/2

h (2ax + b)pax(ax + b) − b2ln

a√

x +pa(ax + b)

i

(28)

Z

p

x3(ax + b) dx =

 b 12a − b

2

8a2x +

x 3

 p

x3(ax + b)+ b

3

8a5/2 ln

a√

x +pa(ax + b)

(29)

x2± a2 dx = 1

2x

√

x2± a2± 1

2a

2ln

x +√

x2± a2

a2− x2 dx = 1

2x

√

a2− x2+ 1

2a

2tan−1 √ x

a2− x2

(31)

Z

x√

x2± a2 dx = 1

3 x

2± a2
3/2

(32)

Z

1

√

x2 ± a2 dx = ln

x +√

x2± a2

(33)

√

a2− x2 dx = sin−1 x

a

(34)

Z

x

√

x2± a2 dx =√

x2± a2

(35)

√

a2− x2 dx = −√

a2− x2

(36)

√

x2± a2 dx = 1

2x

√

x2± a2∓1

2a

2

ln

x +√

x2± a2

(37)

ax2+ bx + c dx = b + 2ax

4a

√

ax2+ bx + c+4ac − b

2

8a3/2 ln

2ax + b + 2pa(ax2+ bx+c)

Z

x√

ax2+ bx + c dx = 1

48a5/2



2√

a√

ax2+ bx + c −3b2+ 2abx + 8a(c + ax2)

+3(b3− 4abc) ln

b + 2ax + 2√

a√

ax2+ bx + c

 (38)

(39) √ 1

ax2+ bx + c dx =

1

√

aln

2ax + b + 2pa(ax2 + bx + c)

(40)

Z

x

√

ax2+ bx + c dx =

1 a

√

ax2+ bx + c− b

2a3/2 ln

2ax + b + 2pa(ax2+ bx + c)

(41)

(a2+ x2)3/2 = x

a2√

a2+ x2

Integrals with Logarithms

(42)

Z

ln ax dx = x ln ax − x

(43)

Z

x ln x dx = 1

2x

2ln x − x

2

4

(44)

Z

x2ln x dx = 1

3x

3

ln x −x

3

9

(45)

Z

xnln x dx = xn+1



ln x

(n + 1)2

 , n 6= −1

(46)

Z ln ax

1

2(ln ax)

2

(47)

Z

ln x

x2 dx = −1

x −ln x x

(48)

Z ln(ax + b) dx =



x + b a

 ln(ax + b) − x, a 6= 0

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

a − 2x

(50)

Z ln(x2− a2) dx = x ln(x2 − a2) + a lnx + a

x − a − 2x

(51)

Z

ln ax2+ bx + c
dx = 1

a

√ 4ac − b2tan−1√2ax + b

4ac − b2−2x+ b

2a + x



ln ax2+ bx + c

(52)

Z

x ln(ax + b) dx = bx

2a − 1

4x

2+ 1 2



x2− b

2

a2

 ln(ax + b)

(53)

Z

x ln a2− b2x2

dx = −1

2x

2 +1 2



x2− a

2

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 = x

2

4 +

1

2x

2

(ln x)2− 1

2x

2

ln x

(57)

Z

x2(ln x)2 dx = 2x

3

27 +

1

3x

3(ln x)2− 2

9x

3ln x

Integrals with Exponentials

(58)

Z

eax dx = 1

ae

ax

(59)

xeax dx = 1

a

√

xeax+ i

√ π 2a3/2erf i√

ax
, where erf(x) = √2

π

Z x

0

e−t2dt

(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 = x

neax

a

Z

xn−1eaxdx

(66)

Z

xneax dx = (−1)

n

an+1 Γ[1 + n, −ax], where Γ(a, x) =

Z ∞

x

ta−1e−tdt

(67)

Z

eax2 dx = −i

√ π

2√

aerf ix

√

a

Z

e−ax2 dx =

√ π

2√

aerf x

√

a

(69)

Z

xe−ax2 dx = − 1

2ae

−ax 2

(70)

Z

x2e−ax2 dx = 1

4

r π

a3erf(x√

a) − x 2ae

−ax 2

Integrals with Trigonometric Functions

(71)

Z sin ax dx = −1

acos ax

(72)

Z sin2ax dx = x

2 − sin 2ax

4a

(73)

Z sin3ax dx = −3 cos ax

cos 3ax 12a

(74)

Z

sinnax dx = −1

acos ax 2F1

 1

2,

1 − n

2 ,

3

2, cos

2ax



(75)

Z cos ax dx = 1

asin ax

(76)

Z cos2ax dx = x

2 +

sin 2ax 4a

(77)

Z cos3axdx = 3 sin ax

sin 3ax 12a

(78) cospaxdx = − 1

a(1 + p)cos

1+pax ×2F1 1 + p

2 ,

1

2,

3 + p

2 , cos

2ax

(79)

Z

cos x sin x dx = 1

2sin

2

x + c1 = −1

2cos

2

x + c2 = −1

4cos 2x + c3

(80)

Z

cos ax sin bx dx = cos[(a − b)x]

2(a − b) − cos[(a + b)x]

2(a + b) , a 6= b

(81)

Z

sin2ax cos bx dx = −sin[(2a − b)x]

4(2a − b) +

sin bx 2b − sin[(2a + b)x]

4(2a + b)

(82)

Z sin2x cos x dx = 1

3sin

3

x

(83)

Z

cos2ax sin bx dx = cos[(2a − b)x]

4(2a − b) − cos bx

2b − cos[(2a + b)x]

4(2a + b)

(84)

Z cos2ax sin ax dx = − 1

3acos

3

ax

(85)

Z

sin2ax cos2bxdx = 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 sin2ax cos2ax dx = x

8 − sin 4ax 32a

(87)

Z tan ax dx = −1

aln cos ax

(88) tan2ax dx = −x + 1

atan ax

(89)

Z

tannax dx = tan

n+1ax a(1 + n) ×2F1 n + 1

2 , 1,

n + 3

2 , − tan

2ax



(90)

Z tan3axdx = 1

aln cos ax +

1 2asec

2ax

(91)

Z sec x dx = ln | sec x + tan x| = 2 tanh−1tanx

2



(92)

Z sec2ax dx = 1

atan ax

(93)

Z sec3x dx = 1

2sec x tan x +

1

2ln | sec x + tan x|

(94)

Z sec x tan x dx = sec x

(95)

Z sec2x tan x dx = 1

2sec

2x

(96)

Z secnx tan x dx = 1

n sec

nx, n 6= 0

(97)

Z csc x dx = ln

tanx 2

= ln | csc x − cot x| + C

(98) csc2ax dx = −1

acot ax

(99)

Z csc3x dx = −1

2cot x csc x +

1

2ln | csc x − cot x|

(100)

Z cscnx cot x dx = −1

n csc

nx, n 6= 0

(101)

Z sec x csc x dx = ln | tan x|

Products of Trigonometric Functions and Monomials

(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

x2cos x dx = 2x cos x + x2 − 2
sin x

(105)

Z

x2cos ax dx = 2x cos ax

2x2− 2

a3 sin ax

(106)

Z

xncos xdx = −1

2(i)

n+1[Γ(n + 1, −ix) + (−1)nΓ(n + 1, ix)]

(107)

Z

xncos ax dx = 1

2(ia)

1−n[(−1)nΓ(n + 1, −iax) − Γ(n + 1, ixa)]

(108) x sin x dx = −x cos x + sin x

(109)

Z

x sin ax dx = −x cos ax

sin ax

a2

(110)

Z

x2sin x dx = 2 − x2
cos x + 2x sin x

(111)

Z

x2sin ax dx = 2 − a

2x2

a3 cos ax + 2x sin ax

a2

(112)

Z

xnsin x dx = −1

2(i)

n

[Γ(n + 1, −ix) − (−1)nΓ(n + 1, −ix)]

(113)

Z

x cos2x dx = x

2

4 +

1

8cos 2x +

1

4x sin 2x

(114)

Z

x sin2x dx = x

2

4 − 1

8cos 2x −

1

4x sin 2x

(115)

Z

x tan2x dx = −x

2

2 + ln cos x + x tan x

(116)

Z

x sec2x dx = ln cos x + x tan x

Products of Trigonometric Functions and Exponentials

(117)

Z

exsin x dx = 1

2e

x(sin x − cos x)

(118)

Z

ebxsin ax dx = 1

a2 + b2ebx(b sin ax − a cos ax)

(119)

Z

excos x dx = 1

2e

x

(sin x + cos x)

(120)

Z

ebxcos ax dx = 1

a2+ b2ebx(a sin ax + b cos ax)

(121)

Z

xexsin x dx = 1

2e

x

(cos x − x cos x + x sin x)

(122)

Z

xexcos x dx = 1

2e

x(x cos x − sin x + x sin x)

Integrals of Hyperbolic Functions

(123)

Z cosh ax dx = 1

asinh ax

(124)

Z

eaxcosh bx dx =







eax

a2− b2[a cosh bx − b sinh bx] a 6= b

e2ax

4a +

x

(125)

Z sinh ax dx = 1

acosh ax

Z

eaxsinh bx dx =







eax

a2− b2[−b cosh bx + a sinh bx] a 6= b

e2ax

4a −x

(127)

Z tanh ax dx = 1

aln cosh ax

(128)

Z

eaxtanh bx dx =















e(a+2b)x

(a + 2b)2F1

h

1 + a 2b, 1, 2 +

a 2b, −e

2bxi

−1

ae

ax

2F1

h

1, a 2b, 1 +

a 2b, −e

2bxi

a 6= b

eax− 2 tan−1[eax]

(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]

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