Integrating sin squared and cos cubed etc

Integrating sin²x and cos²x

(Any Even Power)

Use Cos(2x) formula

cos²x-sinx²=cos2x

1-2sin²x=cos2x

2cos²x-1=cos2x

Rearrange and Integrate as usual

Integrating sin³x and cos³x

(Any odd Power)

Seperate into a product of an even power and single power

i.e

sin²x(sinx)=sin³x

USE C2 trig identity to rearrange the term with power 2

cos²x=1-sin²x

sin²x=1-cos²x

Integrate by SUBSTITUTION using

u = cos(x) or u=sin(x)

Which should give answer in form

u-(1/3)u³