The following wave equation describes a particle in a 1D box that is indfinitely deep (where a is the length of the box)

$\psi (x) = Nx(a-x) e^{i \pi x / a}$ where $ 0 \leq x \leq a$ and

$\psi (x) = 0 $ otherwise

I have already computed the normalization constant N. Now I want to compute the probability to measure the particle's energy in it's ground state. So far, I have:

$p_1 = |c_1|^2 = | < \phi_1 | \psi > |^2 = \bigg( \int_0^a \sqrt{ \frac{2}{a}} \sin(\frac{\pi x}{a}) N x (a - x) e^{i \pi x / a} \bigg)^2$

However, I cant find a way to solve this integral, but I assume there should be some kind of trick that I could use?

Furthermore, how can I determine whether $\psi$ is an eigenstate of the time-independant Schrödinger equation?