You're right! Using your sub we have:
Now simply substitute it all in. Can you go on from here?
I've tried this problem several times and end up with really crazy answers which are too embarrassing to state here. It's:
∫ cos(θ) sin(θ)^43 dθ
I tried with u = sinθ, which I'm not sure is right, and even then I can't integrate the resulting function.