I want to find the Fourier series for $$f(x)=x(a-x).$$ Of course I could do integration by parts and find the coefficients that way, but I'm given a hint to integrate the Fourier series expansion of the delta function with appropriate boundary conditions in order to get the result more quickly.
I know that $$\delta(x-a) = \frac{1}{2\pi} \sum_{k=-\infty}^{\infty} e^{ik(x-a)}$$
but I don't see how to integrate this to recover the result... what's the trick?