Let $f(z):=\frac{e^{iz}}{e^z-e^{-z}}$, then compute $I:=\int _0 ^{\infty} \frac{\sin{x}}{e^x-e^{-x}}dx$
I think it is need to use Residue Theorem. I know $I=\frac{1}{2i} \int _{-\infty}^{\infty} f(x)dx$ And, let $\gamma _R^+ :=R+it, \gamma _R^- :=-R+it \; (0\leq t \leq \pi)$,then $\int _{{\gamma_R}^{+-}} f(z)dz \to 0$ as $R\to \infty$
However $0 ,i\pi$ is on the boundary, so I cannot calculate.