I want to evaulate $\displaystyle{ \int_{-\infty}^{\infty} \frac{e^{kx}}{1+e^x} dx, \ k \in \mathbb{R} }$ via the Residue Theorem over the contour $[x=\pm a, y=0, y=2\pi]$ oriented counterclockwise.
Lets define the complex function, $\displaystyle{ f(z) = \frac{e^{kz}}{1+e^z} }$.
The function has singularities at $z_n = i\pi(2n+1), \ n \in \mathbb{Z}$ and thus we have a single singularity within the desired domain at $n=0$, $z_0 = i\pi$.
Any tips on how to proceed?