Find $ \int_0^\infty \frac{x}{x^3 +1} dx$
What cantor should I choose?
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1Welcome to MSE. What's a cantor? – José Carlos Santos Dec 06 '20 at 12:44
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1Georg Cantor should work. – stoic-santiago Dec 06 '20 at 12:46
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1contour?${}{}{}$ – David Mitra Dec 06 '20 at 12:46
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I was trying to use the complex analysis to integrate it. But I could not choose the good $C_R$ line of integration. – Mr. Proof Dec 06 '20 at 12:49
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1You don't need complex analysis for this. – Toby Mak Dec 06 '20 at 12:52
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1Does this answer your question? $\int_{0}^{\infty}\frac{x}{x^3+1}dx$ =?. Here is a solution with complex analysis: integration using residue. Please use Approach0 to search for duplicates. – Toby Mak Dec 06 '20 at 12:52
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1@TobyMak Thank you so much. – Mr. Proof Dec 06 '20 at 13:51
1 Answers
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Contour integration isn't the best approach. It's better to take $x=\tan^{2/3}t$, so the integral is$$\int_0^{\pi/2}\frac23\tan^{1/3}tdt=\frac{\pi}{3}\csc\frac{2\pi}{3}=\frac{2\pi}{3\sqrt{3}}.$$
J.G.
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1@MathGin Using $2\int_0^{\pi/2}\sin^{2a-1}t\cos^{2b-1}tdt=\operatorname{B}(a,,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$, together with $\Gamma(a)\Gamma(1-a)=\pi\csc\pi a$. – J.G. Dec 06 '20 at 14:11