$$\int_0^{\infty}\frac{1}{1+x^u}\,dx = \frac{1}{u}\Gamma\left(\frac{1}{u}\right)\Gamma\left(1-\frac{1}{u}\right)$$
How to prove this equality and what is the name of it?
2026-04-05 17:34:21.1775410461
How to prove this equality with respect to Gamma function?
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Let $z:=\frac{1}{x^u+1}$, then $x^u=\frac{1-z}{z}$. Rewrite the integral in terms of $z$.