Integral Table 3

Table of Integrals

∫

(a + bx) n dx = dx

(a + bx) n +1 b(n + 1)

for n ≠ −1

1

∫ (a + bx) = b ln a + bx xdx 1 ∫ (a + bx) = b [a + bx − a ln a + bx ] 2

dx

∫ (a + bx)

2

=−

2

=

xdx

∫ (a + bx) dx

∫ (a + bx) ∫

1/ 2

1 b(a + bx)

1 b2

 a  ln(a + bx) +  (a + bx)  

2 = (a + bx) 1 / 2 b

 xdx 2  (a + bx) 3 / 2 = − a(a + bx) 1 / 2  1/ 2 2  3 (a + bx) b   3/ 2

=

−2 b(a + bx) 1 / 2

3/ 2

=

 2  a (a + bx) 1 / 2 + 2  1/ 2  b  (a + bx) 

dx

∫ (a + bx) xdx

∫ (a + bx)

∫ (a ∫ (a ∫ (a ∫ (a ∫ (a ∫ (a ∫ (a ∫ (a

2

dx 1 x = tan −1   + x2 ) a a

xdx 1 = ln(a 2 + x 2 ) + x2 ) 2

2

2

2

2

1 dx x x tan −1   = + + x 2 ) 2 2a 2 ( a 2 + x 2 ) 2a 3 a xdx 1 =− 2 2 2 +x ) 2( a + x 2 ) 1 dx a+x ln = 2 − x ) 2a a − x

xdx 1 = − ln a 2 − x 2 2 2 −x )

2

2

2

1 dx x a+x = 2 2 + 3 ln 2 2 2 a−x 2a ( a − x ) 4a −x ) xdx 1 = 2 2 2 −x ) 2(a − x 2 )

dx

∫

x2 + a2 xdx

∫

2

x +a

2

x 2 dx

∫

x2 + a2

x 3 dx

∫

x2 + a2

∫x

= x2 + a2

=

x x2 + a2 a2 − ln( x + x 2 + a 2 ) 2 2

=

(x 2 + a 2 )3/ 2 − a2 x2 + a2 3

dx

∫x ∫x

 x = ln( x + x 2 + a 2 ) = sinh −1   a

x2 + a2

dx 2

x2 + a2 dx

3

x2 + a2

=−

1  a + x 2 + a 2 ln a  x 

=−

=−

   

x2 + a2 a2x x2 + a2 2

2a x

2

+

 a + x2 + a2 ln  x 2a  1

3

   

x x2 + a2 a2 + ln( x + x 2 + a 2 ) 2 2

∫

x 2 + a 2 dx =

∫

x x 2 + a 2 dx =

∫

x 2 x 2 + a 2 dx =

x( x 2 + a 2 ) 3 / 2 a 2 x x 2 + a 2 a 4 − − ln( x + x 2 + a 2 ) 4 8 8

∫

x 3 x 2 + a 2 dx =

(x 2 + a 2 )5 / 2 a 2 (x 2 + a 2 )3/ 2 − 5 3

(x 2 + a 2 )3/ 2 3

∫

 a + x2 + a2 x2 + a2 dx = x 2 + a 2 − a ln  x x 

∫

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

∫

1  a + x 2 + a 2 x2 + a2 x2 + a2 = − − ln dx x 2a  2x 2 x3 

∫ (x ∫ (x

2

2

dx x = 2 3/ 2 2 +a ) a x2 + a2 xdx = + a 2 )3/ 2

−1 2

x + a2

   

   

∫

x 2 dx = (x 2 + a 2 )3 / 2

∫

x 3 dx = x2 + a2 + (x 2 + a 2 )3 / 2

∫ x( x

−x 2

x +a

2

+ ln( x + x 2 + a 2 ) a2 x2 + a2

1 1  a + x2 + a2 dx = − 3 ln 2 3/ 2  x +a ) a2 x2 + a2 a 

2

   

dx x2 + a2 x = − − 2 2 2 3/ 2 4 4 x (x + a ) a x a x2 + a2

∫

∫x

3

 a + x2 + a2 3 3 −1 dx = − + 5 ln 2 3/ 2  x (x + a ) 2 a 2 x 2 x 2 + a 2 2 a 4 x 2 + a 2 2a  2

1

∫ sin(ax)dx = − a cos(ax)

∫ sin

2

(ax)dx =

x sin( 2ax) − 2 4a

1

∫ cos(ax)dx = a sin(ax)

∫ cos

2

(ax)dx =

x sin(2ax) + 2 4a 1

∫ sin(ax) cos(ax)dx = 2a sin ∫

sin( ax) cos 2 (ax)dx = −

∫

sin 2 (ax) cos(ax)dx =

∫ sin

2

2

(ax)

cos 3 (ax) 3a

sin 3 (ax) 3a

(ax) cos 2 (ax)dx = −

x 1 sin( 4ax) + 32a 8

∫

x sin 2 (ax)dx =

x 2 x sin( 2ax ) cos( 2ax) − − 4 4a 8a 2

∫

x cos 2 (ax )dx =

x 2 x sin( 2ax) cos( 2ax ) + + 4 4a 8a 2

2 2 ∫ x sin ( x)dx =

x 3 x cos(2 x)  x 2 1  − −  −  sin(2 x) 3 4  4 8

∫x

x 3 x cos(2 x)  x 2 1  + +  −  sin(2 x) 3 4  4 8

a

2

cos 2 ( x)dx =

a  mπx   nπx  dx = δ mn  sin  2 a   a 

∫ sin 0

   

a  mπx   nπx   cos dx = δ mn (note: if m = n = 0, then integral = a) 2 a   a 

a

∫ cos 0

∫

a

−a

 mπx   nπx  cos  cos dx = aδ mn (note: if m = n = 0, then integral = 2a)  2a   2 a 

π

1

∫ P (cosθ ) P (cos θ ) sin θdθ = ∫ A

0

∫e

ax

∫ xe

m

PA ( x) Pm ( x)dx =

1 x e a

dx = ax

−1

x 1  dx = e ax  − 2  a a 

2 2x 2  ax  x 2 ax = x e dx e  − 2 + 3 ∫ a  a a 3 3x 2 6 x 6  3 ax ax  x = − + −  x e dx e  ∫ a 2 a3 a4  a

∫x

∫

∞

∫

∞

∫

∞

∫

∞

∫

∞

∫

∞

∫

∞

∫

∞

0

0

0

0

0

0

0

0

n

e ax dx =

x n e ax n n −1 ax − ∫ x e dx a a

x n e − x / α dx = n !α n +1

π

2 2

e − a x dx =

2a

2 2

x e− a x dx =

1 2a 2

π

2 2

x 2 e − a x dx = 2 2

x 3 e − a x dx = 2 2

x 4 e − a x dx =

4a 3 1 2a 4 3 π 8a 5

2 2

x 2 n +1e − a x dx = 2 2

x 2 n e − a x dx =

n! 2a 2 n + 2

[1× 3 × 5 × ... × (2n − 1)] 2

n +1

a

2 n +1

π

2 δ Am 2A + 1