The question is as follows
\begin{align} u_{tt} - u_{xx} &= 0, \quad 0 < x < \pi, t > 0 \\ u(x,0) &= \sin^3(x) \\ u_t(x,0) &= 0 \\ u(0,t) &= u(\pi,t) = 0 \end{align}
After following the standard algorithm, I get
$$E_m = \frac{2}{\pi}\int_{0}^{\pi}\sin^3(x)\sin(mx)dx$$
Now I know from the answer that the solution only exists from $m=1,3$. However, I don't know how to get the integral into a form to see this clearly.