Question

What is the integral of `sin(2x)cos^2(2x)dx`?

Answer 1

`int sin(2x)*cos^2(2x) d x=-1/6cos^3(2x)+C`

Explanation:

`int sin(2x)*cos^2(2x) d x=?`

`"Substitute "u=cos(2x)" ; "d u=-2sin(2x)*d x`

`int sin(2x)*cos^2(2x) d x=-1/2 int u^2*d u`

`"So ;" int u^n d u=1/(n+1)u^(n+1)`

`"Apply "n=2`

`int sin(2x)*cos^2(2x) d x=-1/2*1/3 u^3`

`"Now ;undo substitution"`

`int sin(2x)*cos^2(2x) d x=-1/6cos^3(2x)+C`