How to integrate cosθ*sin(mθ)

⎰ cosθ*sin(mθ) dθ, where m = 1, 2, 3, ...

I've been trying to do this and I can't figure it out.

∫ cos θ sin(mθ) dθ
= ∫ (1/2) [sin(mθ + θ) + sin(mθ - θ)] dθ, via sum/difference of angles formulas
= ∫ (1/2) [sin((m+1) θ) + sin((m-1) θ)] dθ.

If m = 1, then we have ∫ (1/2) [sin(2θ) + 0] dθ = (-1/4) cos(2θ) + C

Otherwise, the answer is
(1/2) [-cos((m+1) θ)/(m+1) - cos((m-1) θ)/(m-1)] + C.

I hope this helps!