Boundary Values & Fourier Series · Challenge problem

A rod $0 \le x \le \pi$ obeys $u_t = u_{xx}$ with $u(0,t) = u(\pi,t) = 0$ and $u(x, 0) = 1$. Find the total exposure at the midpoint, $\displaystyle\int_0^{\infty} u\!\left(\dfrac\pi2, t\right)dt$.