How to compute that $\int_{-\infty}^{\infty}x\exp(-\vert x\vert) \sin(ax)\,dx$

Question

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

Answer 1

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

Answer 2

Since the function is symmetric, $I(a) =\int_{-\infty}^{\infty}x\exp(-\vert x\vert) \sin(ax)\,dx =2\int_{0}^{\infty}x\exp(-x) \sin(ax)\,dx $.

At this point, I would write $\sin(ax) =Im(e^{iax}) $ so that $I(a) =2Im\int_{0}^{\infty}x\exp(-x) e^{iax}\,dx =2Im\int_{0}^{\infty}x\exp(x(-1+ia))dx $ and try to evaluate $\int_{0}^{\infty}x\exp(-bx)dx $, which should not be too difficult.