Integrate [e^(bx)]/(1+e^x)dx from inf to -inf
Sum of the residues times 2pi equals the integral with a path that encloses the residues. That path includes the complex part of path which is above the x axis and the real part which is a straight line on the x axis. It's a semi circle.
The Attempt at a Solution
So I put this into the complex plain and at (pi)i as one of poles and calculate the residue there. However usually the complex curve part goes to zero but in this case it doesn't. Also I never saw an integral in this form before. I know integrals in the form (x^-a)/(1+x) and trig integrals using the Jorden Lemma. Any help will be appreciated.