Saturday, June 23, 2018

calculus - Prove that $int_0^inftyfrac1{t^x+1}dt = fracpi x csc fracpi x$

I'm stuck on this identity:


$$ \int_0^\infty\frac1{t^x+1}dt = \frac\pi x \csc \frac\pi x $$


Could someone show me a proof for this?


What I've tried:


I've thrown a bunch of substitutions and integration by parts at this, but they haven't led me to the answer. I did, however, find these identites: $$ \int_0^\infty\frac1{t^x+1}dt = \int_0^\infty\frac{t^{x-2}}{t^x+1}dt = \int_0^1\left( \frac{1-t}t \right)^{1/x}dt $$ But none of these seem to lead anywhere helpful.


I also tried introducing another variable to turn it into something similar to the Laplace Transform, but I'm not very familiar with methods like that, so they've led nowhere.

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