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Renascence_5.

Integration and series Zealot

Hey, I'm a chinese junior student and prefer solving any kind of integrals and series with real methods.

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$\displaystyle \int_{0}^{1}x^{6}\left [ \left ( 1-\frac{x}{2}\ln\frac{1+x}{1-x} \right )^{2}+\frac{\pi ^{2}x^{2}}{4} \right ]^{-1} \mathrm{d}x=\frac{22548}{336875}$

$\displaystyle \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n-1}\tilde{H}_nH_n}{n^{2}}=\frac{29\pi ^{4}}{1440}+\frac{\pi ^{2}\ln^22}{8}-\frac{\ln^42}{8}-3\mathrm{Li}_4\left ( \frac{1}{2} \right )$

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