Boundary Values & Fourier Series · Hard problem

A rod $0 \le x \le \pi$ obeys $u_t = u_{xx}$ with both ends held at $0$ and initial temperature $u(x, 0) = 2\sin x + 3\sin 3x$. Find the total heat content $\displaystyle\int_0^\pi u(x, \ln 2)\,dx$ at time $t = \ln 2$.