While working on a homework problem related to an infinite potential well, I encountered the following integral: $$ \int_0^L \sin\left(\frac{n\pi}{L}x\right) \sqrt{x(L-x)}\mathrm{d}x $$
This integral is crucial for determining the expansion coefficients of my wavefunction. Despite my efforts, I haven't been able to find an analytical solution. Any guidance or assistance in solving this integral would be greatly appreciated.
EDIT: Pretty important detail I missed is that $n$ is an integer greater than zero.