Nonhomogeneous Equations & Resonance · Challenge problem

Let $y$ solve $y''+2y'+y=2\sin t$ with $y(0)=0$, $y'(0)=0$, and let $y_{ss}$ be the steady-state (periodic) solution of the same equation. Compute $\displaystyle\int_0^{\infty}\big(y(t)-y_{ss}(t)\big)\,dt$.