1) Integrate Cos^n(x) dx

Question

2) Integrate e^(ax)Sinbx dx

3) Integrate (5xCos3x) dx

I Will be happy to critique your thinking on these.

1) Derive a recursive relation.

2) Simplest by replacing sin(bx) by
Exp[i b x] and taking imaginary part at the end.

3)
First integrate sin(yx) and then differentiate the answer w.r.t. y.