I'm reading a book called (Almost) Impossible Integrals, Sums, and Series of Cornel Ioan Vălean
As a solution of two integrals he writes the limits of some $2n$-th derivatives functions: (pag. 14 and pag. 77)
$$\displaystyle\int_{0}^{1}(\ln(1+x)\ln(1-x))^n\mathrm{d}x=\frac{1}{4}\lim_{(x,y)\to(1,1)}\frac{\partial^{2n}}{\partial x^n\partial y^n}2^{x+y}\text{B}(x,y)$$ and $$\displaystyle\int_{0}^{1}\frac{\arctan(x)}{1+x}\ln(x)^{2n}\mathrm{d}x=\frac{\pi}{4}(1-4^{-n})\zeta(2n+1)(2n)!+\frac{(2n)!}{2}\beta(2n+2)+\frac{\pi}{16}\lim_{s\to 0}\frac{\mathrm{d}^{2n}}{\mathrm{d}s^{2n}}\left[\frac{\psi\left(\frac{1-s}{4}\right)-\psi\left(\frac{3-s}{4}\right)}{\sin\left(\frac{\pi s}{2}\right)}+\frac{\psi\left(\frac{2-s}{4}\right)-\psi\left(\frac{4-s}{4}\right)}{\cos\left(\frac{\pi s}{2}\right)}+2\pi\csc(\pi s)\right]$$
Where:
- $\text{B}(x,y)$ is the Beta function;
- $\zeta(z)$ is the Riemann zeta function;
- $\beta(z)$ is the Dirichlet beta function;
- $\psi(z)$ is the digamma function.
How is it possible to calculate those limits?