$\pi$ 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 $0 < a < 1$

π\piπ and Gundermann (happy π\piπ day, Gauss bell)