A student of mine was trying to prove the following formula about the Gamma function:
$$\Gamma(s)\Gamma(1 - s) = \frac{\pi}{\sin(\pi s)}$$
For $0 < s < 1$, this is the same as
$$\int_0^{\infty} e^{-t} t^{s - 1} dt \int_0^{\infty} e^{-t} t^{-s} dt = \int_0^{\infty} \int_0^{\infty} \frac{x^{s-1}y^{-s}}{e^{-x - y}}dxdy$$
After some manipulation she was able to show that this is the same as $$\int_0^{\infty} \frac{x^{s-1}}{x+1} dx$$
and Wolfram Alpha confirms that this is indeed $\frac{\pi}{\sin(\pi s)}$, but we were not able to calculate this integral. Can someone help us?
