$$\int_0^\infty\frac 1{1+x^n}dx=\frac\pi n\csc\left(\frac\pi n\right)$$
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?
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$\begingroup$ 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.) $\endgroup$ – Ian Mar 30 '15 at 19:43
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$\begingroup$ I don't know that. $\endgroup$ – lupin813 Mar 30 '15 at 19:44
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1$\begingroup$ Do a search there are at least 3 question that have this solution. $\endgroup$ – dustin Mar 30 '15 at 19:45
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$\begingroup$ okay. can you solve this one for me as an intro to that? $\endgroup$ – lupin813 Mar 30 '15 at 19:45
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$\begingroup$ 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). $\endgroup$ – Ian Mar 30 '15 at 19:47
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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.
