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I am trying to evaluate $$\int_{0}^{\infty}{\frac{\ln(x)}{x^3-1}dx}$$ I have tried IBP, tried to find a substitution, factorising, and even integrating under the differential sign, all to no avail. I ended up plugging the equation into wolframalpha and it said this integral was equal to $\frac{4\pi^2}{27}$ and I'm not really sure how to come about this answer.

Also as kind of an extension to this problem, I was wondering if there was a way to solve $$\int_{0}^{\infty}{\frac{\ln(x)}{x^n-1}dx}, \quad n\in \mathbb{N}$$

Any help would be appreciated.

ayejay
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  • Do you know any complex analysis? A lot of integrals like that are evaluated with the residue theorem. – Matthew Leingang Nov 08 '22 at 10:43
  • I'm not currently familiar with complex analysis or contour integration, is there any place you suggest I start learning? – ayejay Nov 08 '22 at 10:50
  • Can you use series and trigamma function, if yes, split you integral in to 2 parts, 0 to 1, 1 to infty and use substitution $t=1/x$ for the latter. You will get result in terms of trigamma function and use reflection formula for trigamma function. – OnTheWay Nov 08 '22 at 10:56
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    See also https://math.stackexchange.com/a/4023937/686284 – Quanto Nov 08 '22 at 12:54
  • In general $$\int_{0}^{\infty}{\frac{\ln x}{x^n-1}dx} =\left( \frac\pi n\csc\frac \pi n\right)^2 $$ – Quanto Nov 08 '22 at 23:31

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