Tableof Useful Integrals PDF

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Table of Useful Integrals, etc. 1

∞

∫e

− ax 2

0

∞

1 ฀฀π ฀฀ 2 dx = ฀฀ ฀฀ 2 ฀฀a ฀฀

− ax 2

∫ xe 0

1

∞ 2 − ax 2

∫x e 0

2n − ax 2

∫x e 0

n − ax

3 − ax 2

∫x e

dx =

0

dx =

0

(

)

1

1⋅ 3⋅ 5 ⋅ ⋅ ⋅ 2n − 1 ฀฀π ฀฀ 2 dx = ฀฀ 2 n+1 a n ฀฀ ฀฀a ฀฀

∞

∫x e

∞

1 ฀฀π ฀฀ 2 dx = 4a ฀฀ ฀฀a ฀฀ ฀฀

∞

dx =

1 2a

1 2a 2

∞

∫x

2n+1 − ax 2

e

0

dx =

n! 2a n+1

n! a n+1

Integration by Parts: b

b b

∫UdV = ฀฀฀UV ฀฀฀ a− ฀฀∫VdU a

U and V are functions of x. Integrate from x = a to x = b

a

1

∫sin (ax )dx = − a cos ( ax ) x

∫sin (ax ) dx = 2 − 2

( )

sin 2ax 4a

1

1

∫sin (ax ) dx = − a cos ( ax ) + 3a cos ( ax ) 3

3

( )

( )

( )

3 3x 3sin 2ax sin ax cos ax ∫sin ax dx = 8 − 16a − 4a 4

( )

( (

) )

( (

) )

( ) ( )

฀฀ ฀฀ ฀฀ sin ฀฀ ฀ a − b x฀฀฀− sin ฀ a + b x฀฀฀ where a 2 ≠ b2 2 a+b 2 a−b

( ) ( )

฀฀ ฀฀ sin ฀฀ ฀ a − b x ฀฀ ฀ + sin ฀ a + b x ฀฀ 2 a+b 2 a−b

∫sin ax sin bx dx =

∫cos ax cos bx dx =

( (

) )

( (

) )

(

( ) ( ) 2

(

( )

2 2 ∫ x sin ax dx =

( )

( (

) )

(

)

(

)

(

( ) − x cos( ax)

a2

a

( ) + cos ( ax )

x sin ax

( )

∫cos (ax ) dx =

( )

sin ax a x

2

a2

a

∫cos (ax ) dx = 2 +

( )

2 2 ∫ x cos ax dx =

( )

sin 2ax 4a

Taylor Series: ( n) xo n= ∞ f

( )

( )

eax

(a

2

(x−x )

Geometric Series: ∞ 1 ∑ xn = 1 − x n=0

( )

x cos 2ax 1 ฀฀ x 3 ฀฀x 2 + ฀฀ − 3 ฀ ฀ sin 2ax+ 6 ฀฀4a 8a ฀฀ 4a 2

2

− ax ∫cos bx e dx =

o

(

)

(

)

฀฀ ฀฀ ฀฀฀฀ ฀฀฀฀ cos ฀฀ ฀ a − b x ฀฀฀− cos ฀ a + b x฀฀฀+ x sin ฀ a − b x ฀฀ ฀ − x sin ฀ a + b x ฀ ฀ 2 2 2 2 2 a−b 2 a−b 2 a+b 2 a+b

sin ax

∫ x cos ax dx =

n!

( )

( )

( ) ( )

n=0

) )

x cos 2ax x 3 ฀฀x 2 1 ฀฀ − ฀฀ − 3 ฀ ฀ sin 2ax− 6 ฀฀4a 8a ฀฀ 4a 2

∫ x sin ax sin bx dx =

∑

)

( )

( )

( )

( (

cos 2ax x 2 x sin 2ax − − 4 4a 8a 2

∫ x sin ax dx =

∫ x sin (ax ) dx =

)

฀฀ ฀฀ − cos ฀฀ ฀ a − b x฀฀ ฀ − cos ฀ a + b x ฀ ฀ 2 a+b 2 a−b

∫sin ax cos bx dx =

( )

( )

฀฀a cos bx + bsin bx ฀฀ + b2 ฀

)

n

)

(

)

Euler’s Formula: eiφ = cos φ + isin φ Quadratic Equation and other higher order polynomials: ax 2 + bx + c = 0 x=

−b ± b2 − 4ac 2a

ax 4 + bx 2 + c = 0 ฀฀−b ± b2 − 4ac ฀฀ x = ± ฀฀ ฀฀ ฀฀ ฀฀ 2a ฀฀ ฀฀ General Solution for a Second Order Homogeneous Differential Equation with Constant Coefficients: If: y ฀′฀′+ py ฀′+ qy = 0 Assume a solution for y: y = esx y ฀′= sesx y ฀′฀′= s 2 esx ∴ s 2 esx + psesx + qesx = 0 and

s 2 + ps + q = 0

Hence

sx

s x

y = c1e 1 + c2 e 2

Conversions from spherical polar coordinates into Cartesian coordinates:

x = r sin θ cos φ y = r sinθ sin φ x = r cos θ dv = r 2 sinθ drdθ dφ 0...