Systems of Differential Equations · Superchallenge problem

Let $\mathbf x'=A\mathbf x$ with $A=\begin{pmatrix}-2&1&0\\1&-2&1\\0&1&-2\end{pmatrix}$ and $\mathbf x(0)=(1,0,0)$. Evaluate $\displaystyle\int_0^{\infty}t\,x_1(t)\,dt$. Express your answer as a common fraction.