Sine Cosine Integration Simplification

Question

Given integrals in the form:

A: $\int_{0}^{\pi}sin^2(x/2)sin(x)cos^n(x), n \in\{1,3,5\}$

B: $\int_{0}^{\pi}sin(x/2)cos(x/2)cos^n(x), n \in\{2,4,6\}$

Is there some trick to reduce the complexity of the integrals so calculating by hand becomes viable?

Answer 1

Yes. Use $\sin(\frac x 2) \cos(\frac x 2)=\frac 1 2 \sin \,x$ and $\sin^{2} {(\frac x 2)}=\frac 1 2 (1-cos\, x)$. Then make the substitution $y=\cos\, x$. (Split the integral in A) into two terms before making the substitution).