$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$.
It's easier when either m or n is odd take $ n =2k+1$.
Searching my (very) old notes, I found a general reduction formula I used $$I(m,n)= \int {\sin^m(x) \cos^n(x) }~dx=\frac{\sin ^{m-1}(x)~ \cos ^{n-1}(x)}{m+n}\Big(\sin ^2(x)-\frac{n-1}{m+n-2} \Big)+ \frac{(m-1)(n-1)}{(m+n)(m+n-2)}I(m-2,n-2)$$ After a time, you are then left with simple integrands.