Question

How do you integrate?

`intsin(cos^2 2x)*sin4x dx`

Answer 1

`I=1/2cos(1/2+1/2cos(4x))+C`

Explanation:

We want to integrate

`I=intsin(cos^2(2x))*sin(4x)dx`

Remember the identity

`color(blue)(cos^2(a)=1/2+1/2cos(2a)`

Thus

`I=intsin(1/2+1/2cos(4x))*sin(4x)dx`

Make a substitution `u=1/2+1/2cos(4x)=>du=-2sin(4x)dx`

`I=-1/2intsin(u)du`

`I=1/2cos(u)+C`

Substitute back `u=1/2+1/2cos(4x)`

`I=1/2cos(1/2+1/2cos(4x))+C`