TITLE: Evaluate $ \int_{0}^{1} \int_{0}^{1} \frac{1}{1 - (xy)^2}\, dx \, dy $
QUESTION [5 upvotes]: Evaluate:
$$ \int_{0}^{1} \int_{0}^{1} \frac{1}{1 - (xy)^2}\, dx \, dy $$
Can someone provide a hint to tackle this problem? I only know how to do substitution and integration by parts for one variable. My difficulty so far is to express the term $ \dfrac{1}{1 - (xy)^2} $ in term of $ x $ treating $ y $ as a constant so that I can use substitution.

REPLY [6 votes]: The section “Three times $\pi^2/6$” in
Proofs from THE BOOK
by Aigner & Ziegler shows how to evaluate this integral using the completely non-obvious substitution
$$
x = \frac{\sin u}{\cos v}
,\qquad
y = \frac{\sin v}{\cos u}
,
$$
which magically turns the integral into
$$
\iint_E dudv
,
$$
where $E$ is the triangle with corners in $(u,v)=(0,0)$, $(\pi/2,0)$ and $(0,\pi/2)$.
This is a special case of the evaluation of $\sum_{k\ge 0} (-1)^{nk} (2k+1)^{-n}$ in a paper by Beukers, Calabi and Kolk.
(See also this question.)