# How to evaluate the integral $\int_0^\infty \frac{\sin(x)}{\sinh(x)}dx$ with residue theorem [duplicate]

I'm sitting and having fun with contour integration, and I managed to evaluate the integral, $$\int_0^\infty \frac{\cos(x)}{\cosh(x)}dx,$$ with inspiration from: Evaluating $\int_{-\infty}^{\infty}\frac{\cos x}{e^x + e^{-x}}$ using the Residue Theorem. Now, I want to evaluate the following integral, $$\int_0^\infty \frac{\sin(x)}{\sinh(x)}dx,$$ but this seems to be more difficult. Can I do the contour integration over a rectangle as provided in the link above? I hope someone can help me to get started.

• What is important in the fact you are sitting ? – Jean Marie Jun 10 '20 at 16:39