To prove the energy conservation law, we have the time derivative of $E(t)$. So:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ u_{tt}^*u_t + u_t^*u_{tt} + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We make use of the wave equation to rewrite the equation:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ \left( c^2u_{xx} \right)^*u_t + u_t^*\left( c^2 u_{xx} \right) + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We then notice that these terms can be combined as a derivative, since $u_tu_{xx} + u_xu_{xt} = \frac{d}{dx}u_tu_x$ So:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \int_0^{\infty} \left[ \frac{d}{dx} \left( u_x^*u_t \right) + \frac{d}{dx} \left( u_t^*u_x \right) \right] dx
\end{equation}
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \left( u_x^*u_t |_0^{\infty} + u_t^*u_x |_0^{\infty} \right)
\end{equation}
We neglect the terms at $\infty$, since we assume the function is fast decaying. We then make use of the boundary condition to rewrite this:
\begin{equation}
\frac{\partial E(t)}{\partial t} = -\frac{c^2}{2} \left( -i \alpha u_t^*u_t + i \alpha u_t u_t^* \right) = 0
\end{equation}
Thus we have proven the energy conservation law.
