Integration Formulas

Integration Formulas Note that a and b are constants. Elementary Forms ∫ a dx = ax ∫(u +v ) dx = ∫u dx +∫v dx ∫ u dv = uv − ∫ v du dv

du

∫ u dx dx = uv − ∫ v dx dx

∫x

dx =

n

x n +1 n +1

n ≠ −1

∫

dx = ln x x

∫

df ( x ) dx dx = ln( f ( x ) ) f ( x)

∫b

ax

dx =

b ax a ln b

Exponential Forms ∫ e dx = e x

∫e

−x

∫e

ax

x

dx = −e −x

dx =

e ax a

b >0

∫ xe

ax

dx =

e ax ( ax −1) a2

Logarithm Forms ∫ln x dx = x ln x − x

∫ x ln x dx = ∫x

2

ln x dx =

∫(ln x )

∫

x2 ln x − 14 x 2 2

2

x3 ln x − 19 x 3 3

dx = x (ln x ) −2 x ln x +2 x 2

( ln x ) n

1 ( ln x ) n +1 n +1

dx =

x

dx

∫ x ln x = ln( ln x )

Forms with a+bx ( a + bx ) n ∫ ( a + bx ) dx = ( n +1)b

n +1

dx

n ≠ −1

1

∫ a + bx = b ln( a + bx ) dx

∫ a + bx

=

2

dx

∫ ( a + bx )

2

=−

1 b( a + bx )

3

=−

1 2 2b( a + bx )

dx

∫ ( a + bx )

 x ab  1  tan −1   a  ab  

ab > 0

∫x

2

dx 1  x −a  = ln   2 −a 2a  x + a 

∫ ( a + bx ) x dx

∫ a + bx

dx =

=

x dx

∫ a + bx

n

2

2

( a + bx ) n +1

n ≠ −1

( n +1)b

1 ( a + bx − a ln(a + bx ) ) b2

=

x dx

∫ ( a + bx )

x2 > a2

1 ln (a + bx 2 ) 2b

=

1  a  ln ( a + bx ) +  b2  a + bx 

x dx 1 ∫ ( a + bx ) = b ( ( a + bx ) 2

1 2

3

x 2 dx

∫ ( a + bx ) ∫

2

=

1 b3

a + bx dx =

∫x ∫x

2

∫

a 2 − x 2 dx =

∫x

∫

)

( a + bx ) 3

2 ( 3bx − 2a ) 15b 2

a + bx dx =

( a + bx ) 3

2 (8a 2 −12abx +15b 2 x 2 ) 105b 3

 1  x a 2 − x 2 + a 2 sin −1  x a 2  

a 2 − x 2 dx = −

dx 2 = b a + bx

− 2a ( a + bx ) + a 2 ln ( a + bx )

 a2   a + bx − 2a ln ( a + bx ) − a + bx    

2 3b

a + bx dx =

2

1 3

a + bx

(a

2

− x2 )

3

   

( a +bx ) 3

∫

x dx 2( 2a − bx ) =− 3b 2 a + bx

∫

x 2 dx 2(8a 2 − 4abx + 3b 2 x 2 ) = 15b 3 a + bx

∫

a2 − x2

∫

x2 ± a2

∫x

∫x ∫

dx

= sin −1

=ln x + x 2 ± a 2

dx x2 −a2

dx a2 ± x2

x dx x2 ± a2

a + bx

x a

(

dx

a + bx

)

1 x sec −1 a a

=

=−

1  a + a2 ± x2 ln a  x 

= x2 ± a2

Trigonometric Forms 1

∫ sin ax dx = − a cos ax 1

∫ cos ax dx = a sin ax 1

∫ tan ax dx = − a ln(cos ax ) 1

∫ cot ax dx = a ln(sin ax ) 1

∫ sec ax dx = a ln(sec ax + tan ax ) 1

∫ csc ax dx = a ln(csc ax − cot ax )

   

∫ sin

2

ax dx =

x sin 2ax − 2 4a

cos ax ( 2 + sin 2 ax ) 3a

∫ sin

3

ax dx = −

∫ sin

4

ax dx =

3 x sin 2ax sin 4ax − + 8 4a 32a

ax dx =

x sin 2ax + 2 4a

ax dx =

sin x ( 2 + cos 2 ax ) 3a

ax dx =

3 x sin 2ax sin 4ax + + 8 4a 32a

∫ cos

2

∫ cos

3

∫ cos

4

∫sin ax sin bx dx =

sin ( ( a − b) x ) sin( ( a + b) x ) − 2( a − b ) 2( a + b )

∫ cos ax cos bx dx = ∫ sin ax cos ax dx =

sin ( ( a − b) x ) sin ( ( a + b) x ) + 2( a − b ) 2( a + b )

2

ax cos 2 ax dx =

1

cos( ( a − b) x ) cos( ( a + b) x ) − 2( a − b ) 2( a + b )

x sin 4ax − 8 32a

x cos ax a

∫ x sin ax dx = a

2

∫x

2x a2 x2 −2 sin ax − cos ax 2 a a3

2

sin ax dx =

1

∫ x cos ax dx = a

2

a 2 ≠ b2

sin 2 ax 2a

∫sin ax cos bx dx = − ∫ sin

a 2 ≠ b2

sin ax −

cos ax +

x sin ax a

a 2 ≠ b2

∫x

2

cos ax dx =

2x a 2 x 2 −2 cos ax + sin ax 2 a a3

1 tan ax − x 2a

∫ tan

2

ax dx =

∫ tan

3

ax dx =

1 1 tan 2 ax + ln ( cos ax ) 2a a

∫ tan

4

ax dx =

1 1 tan 3 ax − tan x + x 3a a

dx 1 = − cot ax 2 ax a

∫ sin

dx

∫ cos

∫ sin

2

ax

−1

=

1 tan ax a

ax dx = x sin −1 ax +

1 a

1 −a 2 x 2

∫ cos

−1

ax dx = x cos −1 ax −

1 a

∫ tan

−1

ax dx = x tan −1 ax −

1 ln (1 + a 2 x 2 ) 2a

1 −a 2 x 2

Hyperbolic Trigonometric Forms ∫sinh x dx = cosh x ∫cosh x dx = sinh x ∫ tanh x dx = ln(cosh x ) ∫ x sinh x dx = x cosh x −sinh x ∫ x cosh x dx = x sinh x −cosh x