Qiaochu, Parseval does apply, it just doesn't yield anything elementary. :) Still, it is easy to deduce (for example) the identity
but$$\int_{0}^{2\pi}x(2\pi-x)\log(1-\cos{x})dx=8\pi \zeta(3)-\frac{4}{3}\pi^3\log{2}$$ but its not clear what good such an identity is, relative to the original problem.