MICHAEL PENN´S CHALLENGE INTEGRAL

Last week I came across a video of prof. Michael Penn where he proves the following result.


\begin{align*} \int_0^1\frac{\ln(1-x)}{1+x^2}\,dx&=\frac{\pi \ln(2)}{8}-G \label{1} \end{align*}


Where G is Catalan´s constant. He solved this integral by a smart substitution x \to \tan(x) as we will see in this paper, and left a challenge to solve this integral by another method. After many attempts I finally came up with a solution in terms of Polylogarithms, which is much harder and less intuitive but enabled me to generalize the above integral, namely:


\begin{align*} \int_0^1 \frac{\ln^{s-1}(x)}{1-2x+x^2}\,dx&=\frac{i}{2}(-1)^{s}\Gamma(s)\left[\operatorname{Li_{s}}\left(\frac{1+i}{2} \right)-\operatorname{Li_{s}}\left(\frac{1-i}{2} \right) \right] \end{align*}


Click here to see the proof.

Click here to see another collection of integrals related to Catalan´s constant

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