These questions are all about Fourier analysis.

Question

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 $$\int_0^1\frac{\ln{x}}{1+x}\mathrm dx=-\frac{\pi^2}{12}.$$

Answer 1

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 $$ \begin{align} \int_0^1\frac{\log(x)}{1+x}\mathrm{d}x &=\sum_{k=0}^\infty(-1)^k\int_0^1x^k\log(x)\,\mathrm{d}x\\ &=-\sum_{k=0}^\infty\frac{(-1)^k}{(k+1)^2}\\ &=-\frac{\pi^2}{12} \end{align} $$