Change of variables integration for all real p>1?

Asked Mar 19 '19 at 00:41

Active Mar 19 '19 at 00:51

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Originally I have the following problem to show it holds for any real $p>1$, $$\int_0^\infty \frac{1}{x^p+1}\, \mathrm{d}x = \frac{\pi}{p}\sin\left(\frac{\pi}{p}\right).$$ However, since there are infinitely many $p$ that fit this kind of form, I was thinking of approaching it in another way. Specifically change of variables integration. I'm embarrassed to admit that I do not know how to proceed.

Thinking: $c_\alpha \int_0^\infty \frac{x^\alpha}{x+1} \, \mathrm{d}x$ where $c$ is a constant and we have a single pole at $z=1$.. I do not trust my change of variables. I may not know how to do this...

Any direction on change of variables integration is appreciated. I know after that I would discuss removable singularities and proceed with Residue Theorem and thus ultimately solving my problem.

edited Mar 19 '19 at 00:51

Brian61354270

2,170

asked Mar 19 '19 at 00:41

R.ron

1

May I suppose that the rhs could be $\frac{\pi }{p} \csc \left(\frac{\pi }{p}\right)$ instead ? – Claude Leibovici Mar 19 '19 at 04:45
By setting $n=0,$ this is a special case of Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ – Brevan Ellefsen Mar 19 '19 at 09:15
I don't trust how to do the change of variables that you are suggesting Brevan? – R.ron Mar 19 '19 at 18:33

Change of variables integration for all real p>1?

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