Among the several proofs known for $\sum_{ n=1}^\infty{ 1/n^2} = { \pi^2/6}$, the one by Beukers, Calabi, and Kolk involves the evaluation of $\,\int_0^1 { \int_0^1{ 1/(1-x^2 y^2) \, dx} \, dy}$. It starts by showing that this double integral is equivalent to $\frac34 \sum_{ n=1}^\infty{ 1/n^2}$, and then a non-trivial \emph{ trigonometric} change of variables is applied which transforms that integral into $\,{ \int \int}_T \: 1 \; du \, dv$, where $T$ is a triangular domain whose area is simply ${ \pi^2/8}$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it to the above integral, I find exact closed-form expressions for $\int_0^\infty{ [\sinh^{ -1}{ (\cosh{ u})}-u] d u}$, $\,\int_{ \alpha}^\infty{ [u-\cosh^{ -1}{ (\sinh{ u})}] d u}$, and $\,\int_{ \,\alpha/2}^\infty{ \ln{ (\tanh{ u})} \: d u}$, where $\alpha = \sinh^{ -1}(1)$. From the latter integral, I also derive a two-term dilogarithm identity.
Categories: math.CA math.NT