Stuff like this makes me want to drop out.
$\displaystyle \int\sin^5 (4x) \cos^2 (4x)dx$
I think I would be fine if it weren't for those coefficients in there.
Thanks
Since sine is raise to an odd power, break off a factor of sine:
$\displaystyle \int\sin^5\!\left(4x\right)\cos^2\!\left(4x\right) \,dx=\int\left[\sin^2\!\left(4x\right)\right]^2\cos^2\!\left(4x\right)\sin\!\left(4x\right)\,dx$
Now apply the identity $\displaystyle \sin^2u=1-\cos^2u$ to get:
$\displaystyle \int\left[1-\cos^2\!\left(4x\right)\right]^2\cos^2\!\left(4x\right)\sin\!\left(4x\right)\,dx$.
Now let $\displaystyle u=\cos\!\left(4x\right)\implies -\tfrac{1}{4}\,du=\sin\!\left(4x\right)\,dx$.
Can you take it from here?