Laplace Transforms · Hard problem

Let $y$ solve $y'' + 3y' + 2y = 0$, $y(0) = 1$, $y'(0) = 1$, and let $Y = \mathcal{L}\{y\}$. Using $\displaystyle\int_0^\infty y(t)\,dt = Y(0)$, find $\displaystyle\int_0^\infty y(t)\,dt$ without solving for $y$.