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I am trying to prove the following 2 identities using complex analysis methods and contour integration and I'm really stuck on defining the integration paths.

[tex]\int_{0}^{1}\frac{\log(x+1)}{x^2+1}d x=\frac{\pi\log2}{8}[/tex]

[tex]\int_{0}^{\infty}\frac{x^3}{e^x-1}d x=\frac{\pi^2}{15}[/tex]

It's interesting that the first integral is proper in both limits of integration, whereas the second - improper in both limits.

I am familiar with a proof of the second identity using series, uniform convergence and the gamma function. This proof also verifies the following generalisation of the second identity:

[tex]\int_{0}^{\infty}\frac{x^{s-1}}{e^x-1}d x=\Gamma(s)\zeta(s)[/tex]

But I'm looking for a way in the complex plain.

Any help or hints are much appreciated!

Regards, Marin

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# Contour integral methods

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