For $n\in\mathbb{N}$, I'm trying to find a closed form for the following integrals :
$$I\_n=\int\_{0}^{\infty}e^{-x}\sin(n\ln(x))\text{d}x$$

My real objective is to evaluate $\sum\limits\_{n=1}^{\infty}\frac{I\_n}{n}$, and since interchanging the sum and the integral didn't lead anywhere, I suppose that finding a closed form expression for $I\_n$ is the way to go, but I'm lost as of how to proceed...

Maybe the residue theorem/contour integration could help, but I'm not familiar with complex analysis so I haven't tried it - feel free to use it though.

Any suggestion ?