Laplace Transforms · Superchallenge problem

Let $y$ solve $y'' + 3y' + 2y = e^{-3t}$, $y(0) = 1$, $y'(0) = 0$. Using $\displaystyle\int_0^\infty t\,y(t)\,dt = -Y'(0)$, where $Y = \mathcal{L}\{y\}$, find $\displaystyle\int_0^\infty t\,y(t)\,dt$.