TITLE: Closed form of $\int_{0}^{1}\int_{0}^{1}\frac{1}{1-\frac{xy}{2}}dxdy$
QUESTION [6 upvotes]: I've tried using the Jacobian matrix, but this method yielded no significant results. Another way is to turn this integral into the
$$
\sum_{n=0}^{\infty}\frac{1}{(n+1)^22^n}
$$
However, I do not know how to find the exact sum of this series.

REPLY [3 votes]: As has been mentioned in the comments, the function to use here is 
$$\mathrm{Li}_2(x)=\sum_{k\ge1}\frac{x^k}{k^2}$$
so that your sum is given by $S=\frac12\mathrm{Li}_2(\frac12)$. To evaluate this exactly, we use the formula
$$\mathrm{Li}_2(z)+\mathrm{Li}_2(1-z)=\frac{\pi^2}{6}-\ln(z)\ln(1-z)$$
and plug in $z=1/2$ to get
$$S=\frac{\pi^2}{24}-\frac{\ln^2(2)}{4}.$$