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While going through some of my notes, I came across a formula for evaluating an integral of the form: $$\int_0^\infty {\frac{dx}{x^n+1}} = \frac{\pi}{n\sin(\frac{\pi}{n})}$$

I was wondering if any other formulas exist for solving more complicated integrals where an antiderivative can't easily be found. If you could also include a proof of said formula, that would also be greatly appreciated.

Thanks in Advance!

  • The proof of this formula is very easy with complex integration (https://math.stackexchange.com/q/247866) and very tedious with real integration. – Jean Marie Mar 13 '19 at 02:14
  • @JeanMarie - With a single complex change of variable, or two simple change of variables this can be achieved very EASILY with real based methods. –  Mar 13 '19 at 02:36
  • @DavidG I am interested by a reference to this method. – Jean Marie Mar 13 '19 at 02:44
  • @JeanMarie - I cover a more complicated form here - https://math.stackexchange.com/questions/3057298/solving-used-real-based-methods-int-0x-fractk-lefttn-a-rightm-d –  Mar 13 '19 at 02:45
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    @DavidG Thank you very much ! – Jean Marie Mar 13 '19 at 02:45
  • @JeanMarie - No worries, I hope it's of use :-) –  Mar 13 '19 at 02:46

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