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Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: $$\int_0^1\frac{\ln{x}}{1-x}\mathrm dx=-\frac{\pi^2}{6}$$ and $$\int_0^1\frac{\ln{x}}{1+x}\mathrm dx=-\frac{\pi^2}{12}.$$

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Expanding into series and integrating by parts: $$ \begin{align} \int_0^1\frac{\log(x)}{1-x}\mathrm{d}x &=\sum_{k=0}^\infty\int_0^1x^k\log(x)\,\mathrm{d}x\\ &=-\sum_{k=0}^\infty\frac1{(k+1)^2}\\ &=-\frac{\pi^2}{6} \end{align} $$ and $$ \begin{align} \int_0^1\frac{\log(x)}{1+x}\mathrm{d}x &=\sum_{k=0}^\infty(-1)^k\int_0^1x^k\log(x)\,\mathrm{d}x\\ &=-\sum_{k=0}^\infty\frac{(-1)^k}{(k+1)^2}\\ &=-\frac{\pi^2}{12} \end{align} $$

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