How do we prove such integral? [duplicate]

$$\int_0^\infty\frac 1{1+x^n}dx=\frac\pi n\csc\left(\frac\pi n\right)$$

If we want to prove the left side equal to the right side in this case, how do we start? How do we prove this definite integral?

• Do you know about complex contour integration? This is the approach that I would use for this problem (for $n>2$; for $n=1$ and $n=2$ the problem is pretty easy.) – Ian Mar 30 '15 at 19:43
• I don't know that. – lupin813 Mar 30 '15 at 19:44
• Do a search there are at least 3 question that have this solution. – dustin Mar 30 '15 at 19:45
• okay. can you solve this one for me as an intro to that? – lupin813 Mar 30 '15 at 19:45
• Without contour integration this problem is not easy. If I were told "do this without using contour integration" I would have to partial fraction expand $\frac{1}{1+x^n}$, which still requires complex numbers (albeit not the residue theorem). – Ian Mar 30 '15 at 19:47

1 Answer

Hint: Let $t=\dfrac1{1+x^n}$ , then recognize the expression of the beta function in the new integral,

and employ Euler's reflection formula for the $\Gamma$ function in order to obtain the desired result.