\(\pi\) and Gundermann (happy \(\pi\) day, Gauss bell)

Calculus Level 4

Evaluate: \[\displaystyle \int_{-\infty}^{\infty} \frac{e^{ax}}{e^x + 1} \,dx\] when \(a = \frac{1}{2}\)

Bonus.- Find a beautiful closed form for the above integral being \(0 < a < 1\)