I want to know the proof of the definite integral :

Question

$$ \int _{0}^{\infty }\frac {x^{p-1}dx}{1+x}=\frac {\pi }{\sin(p\pi )}$$ for $0<p<1$.

Please do let me know the proof of this specified integral.

Have you tried using contours in the complex plane? – Gregory Mar 09 '17 at 19:30 — Gregory, Mar 09 '17 at 19:30

score 0 · Answer 1 · answered Mar 09 '17 at 19:52

0

As you can read here , we have the Beta Function, its relation with the Gamma Function and Euler's Reflection Formula:

$$\int_0^\infty\frac{ x^{p-1}}{1+x}dx= B(p,1-p)=\frac{\Gamma(p)\Gamma(1-p)}{\Gamma(1)}=\frac\pi{\sin \pi p}$$

answered Mar 09 '17 at 19:52

DonAntonio

Yeah, i know that this equation corresponds to Euler's Reflection Formula. But In lieu of deducing this specified reflection formula i came across with this particular definite integral and for that matters i want to know the proof of that specified definite integral which is of course a standard result i presume. So, if u know then do comment down bellow. – Sayandip Banerjee Mar 10 '17 at 13:58
@SayandipBanerjee Well, as commented already: perhaps with complex analysis. Do you know that? – DonAntonio Mar 10 '17 at 15:00
I haven't started that though but if u can tell me the approach den i could try solving it..... – Sayandip Banerjee Mar 10 '17 at 15:31

1 Answers1