How do we prove such integral?

Question

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

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
Do a search there are at least 3 question that have this solution. — dustin, 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
Literally all you needed to do was a search: pick your poison — dustin, Mar 30 '15 at 19:51
For posting on this site, you need to learn math markup! ($\LaTeX$) — kjetil b halvorsen, Mar 30 '15 at 19:56

score 0 · Answer 1 · answered Mar 30 '15 at 19:59

0

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.

answered Mar 30 '15 at 19:59

Lucian

1 Answers1