You need to use the identity \cos(2x) = 1 - 2\sin^2(x) to turn \sin^2(x) into a sum of things you know how to integrate (either directly or using integration by parts).
I applied for maths with physics, and switched to straight maths a couple of weeks into my first term. The reason for this was that I really wasn't interested in doing any practicals. I didn't really get much of a chance to gauge the workload, but if your college makes you do all four of the...