How to compute that $\int_{-\infty}^{\infty}x\exp(-\vert x\vert) \sin(ax)\,dx$ $$
\int_{-\infty}^{\infty}x\exp(-\vert x\vert)
\sin(ax)\,dx\quad\mbox{where}\
a\ \mbox{is a positive constant.}
$$
My idea is to use integration by parts. But I have been not handle three terms..
Please, help me solve that.

or note
$$
2\int_{0}^{\infty}x\exp(-x)
\sin(ax)\,dx
=
2\;\mathrm{Im}\int_{0}^{\infty}x\exp((-1+ia)x)\;dx
$$
