[psyang]

Originally Posted by bizaro86

One comment I think is quite useful but isn't a solution.

Spoiler!

I think I can get part of the way there using the hint of no pen/paper as a hint, because there aren't many ways to solve an integral by inspection.

cos(x/2) is an even function. x^3 is an odd function. An even function multiplied by an odd function is odd, so cos(x/2)*x^3 is odd. Any definite integral of an odd function between a and -a is zero, which using integration by parts makes that whole term 0.

So if we distribute it that gets us down to the (1/2)*integral[sqrt(4-x^2)]

I don't think I can solve that piece - I suspect some sort of substitution is required.

Definitely on the right track - keep going!