Boundary Values & Fourier Series · Hard problem

Find the Fourier sine series of $f(x) = x(\pi - x)$ on $[0, \pi]$, and evaluate it at $x = \dfrac\pi2$ to find $\displaystyle\sum_{k=0}^{\infty}\dfrac{(-1)^k}{(2k+1)^3} = 1 - \dfrac1{27} + \dfrac1{125} - \cdots$.