I tried to solve this integral but I have problem with the cubic term because I can't add $\sin(\frac{n{\pi}x}{L})$:
$$\int_{0}^1 \Biggl[ A\Biggl(\sum_{n=1}^N q_n(t)\sin(\frac{n{\pi}x}{L})\Biggr)^3\sin(\frac{m{\pi}x}{L})\Biggr]dx $$
where $n = 1$ to $30$; $m = 1$ to $30$; $A$ is a constant.