A problem I'm currently working on asks to evaluate the following integrals:
$\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx\;$ and $ \ \int_{0}^{\infty}\frac{x\cos(ax)}{\sinh(x)}dx$
I've seen previous questions around here that show how to do the evaluation via contour integration (Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour and show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $) but this particular problem requires that I use the following contour:
I tried applying the previous two methods using this contour, but the final answer I end up with keeps ending up as a completely different form and I am unable to prove that the answer I got is exactly the same as the answers using the different methods linked above.
Any help would be greatly appreciated