I know I am supposed to strip one layer away, in this case from sin, to give me:
But I don't know what to do next, only that it will probably involve u substitution.
You can solve this by parts, but it's tedious.
Pick either the sin or cos part to be u, and the other to be dv. You will end up with a bit you can evaluate and another bit you can integrate by parts again.
Then you integrate by parts again and you get another integration with reduced indices.
You will be able to deduce a rule which goes something like this (from my book of integrals):
If you change x ---> u=cos(x)
The boundaries of the integrals *have to* be the ones for u, not for x.
If x=0 ---> u=...
If x=pi/2 ---> u=...
These are the new boundaries/limits of the integral
So either you change it, either you substitute back u by cos x the integral you will get. But here, it'll be easier to change the boundaries/limits
NB : what you've got so far is correct