Table of Useful Integrals

Table of Useful Integrals, etc.

1

e-ax2 dx

0

=

1 2

a

2

xe-ax2 dx =

1

0

2a

1

0

x2e-ax2 dx

=

1 4a

a

2

0

x3e-ax2 dx

=

1 2a2

1

( ) 0

x e 2n - ax2

dx

=

1

35 2n 2n+1 a n

-

1

a

2

0

x2n+1e- ax2 dx

=

n! 2an+1

0

xne-ax dx

=

n! an+1

Integration by Parts:

b

b

b

UdV = UV a - VdU U and V are functions of x. Integrate from x = a to x = b

a

a

sin

(

ax)

dx

=

-

1 a

cos(ax)

sin2

( ax ) dx

=

x 2

-

sin ( 2ax )

4a

sin3

(ax)

dx

=

-

1 a

cos

(

ax

)

+

1 3a

cos3

(ax)

sin4

(ax)dx

=

3x 8

-

3sin ( 2ax )

16a

-

sin3

(ax)cos(ax)

4a

sin

(

ax)sin

(

bx)

dx

=

( sin a 2(a

- -

b) b)

x

-

( sin 2(

a a

+ +

b) b)

x

where a2 b2

cos ( ax

)cos(bx

)

dx

=

( sin 2(

a a

- -

b) b)

x

+

( sin a 2(a

+ +

b) b)

x

sin ( ax ) cos ( bx ) dx

=

-

cos (a - b) 2(a - b)

x

-

( cos a + 2(a +

)b x b)

( ) ( ) ( ) xsin2

ax

x2 x sin 2ax

dx = -

4

4a

cos 2ax - 8a2

( ) ( ) ( ) x2 sin2

ax

dx =

x3 x2 1 6 - 4a - 8a3 sin

2ax

x cos 2ax - 4a2

x sin(ax)sin(bx)dx

=

( ) cos a - b x

2

(

a

-

)2

b

-

( ) cos a + b x

2

(

a

+

)2

b

+

( ) x sin a - b x

2

(

a

-

)2

b

-

x sin (a + b)

2

(

a

+

)2

b

x

x

sin ( ax )

dx

=

sin ( ax )

a2

-

x

cos(ax)

a

x cos(ax)dx

=

x sin(ax)

a

+

cos(ax)

a2

cos ( ax ) dx

=

sin ( ax )

a

cos2

( ax ) dx

=

x 2

+

sin ( 2ax )

4a

( ) ( ) ( ) x2 cos2

ax

x3 x2 1 dx = 6 + 4a - 8a3 sin

2ax

x cos 2ax + 4a2

( ) ( ) ( ) ( ) cos bx e-ax2 dx =

eax a2 + b2

a cos bx

+ bsin bx

Taylor Series:

f (n)

( ) n=

x o

( ) n=0 n!

n

x

-

x o

Geometric Series:

xn =

1

n=0

1- x

Euler's Formula: ei = cos + i sin

Quadratic Equation and other higher order polynomials: ax2 + bx + c = 0

x = -b ? b2 - 4ac 2a

ax4 + bx2 + c = 0

x

=

?

-b

?

b2 - 4ac

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 = s2esx

s2esx + psesx + qesx = 0

and s2 + ps + q = 0

Hence

y

=

c es1x 1

+

c es2 x 2

Conversions from spherical polar coordinates into Cartesian coordinates:

x = r sin cos y = r sin sin x = r cos dv = r2 sindrdd 0 ................
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