Conservation of wave energy

Differentiate $E(t)$ under the integral sign: $E'(t) = \int_0^L \left(u_t u_{tt} + c^2 u_x u_{xt}\right)dx$.

Substitute $u_{tt}=c^2 u_{xx}$ from the wave equation to get $E'(t) = \int_0^L \left(c^2 u_t u_{xx} + c^2 u_x u_{xt}\right)dx = c^2\int_0^L \left(u_t u_{xx} + u_x u_{xt}\right)dx$. Recognize the integrand as the derivative $\tfrac{\partial}{\partial x}(u_t u_x)$, so $E'(t) = c^2\Big[u_t u_x\Big]_0^L$.

Apply the fixed-end boundary conditions $u(0,t)=u(L,t)=0$, which force $u_t(0,t)=u_t(L,t)=0$ as well (differentiating the boundary conditions with respect to $t$). Therefore $E'(t) = c^2\big(u_t(L,t)u_x(L,t) - u_t(0,t)u_x(0,t)\big) = 0$, so $E(t)=E(0)$ is constant for all $t\ge 0$.