Problem. Evaluate the following integral. \begin{equation}\label{eqn:wts} \int_{0}^{1} \log (1-x) \log x \log (1+x) \; dx \end{equation} We divide the solution into several steps. 1. Reduction to Euler series. The key ingredient for the reduction is the following integral. \begin{equation*} \int_{0}^{\frac{\pi}{2}} \sin^{p} \theta \cos^{q} \theta \log \sin\theta \log \cos\theta \; d\theta. \end{..
Problem. Prove the following identities. \begin{equation}\label{eqn:wts} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}} = \frac{2}{5} \zeta(3), \qquad \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 2^n \binom{2n}{n}} = \frac{1}{4} \zeta(3) - \frac{1}{6}\log^3 2 \end{equation} We divide the proof into several lemmas. Lemma 1. For $|x| < 1$. \begin{equation}\label{eqn:wts_lem_1} \int_{0}^{x} ..
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