I found the following bivariate integral and I do not know how to attack it. $$ \int_0^\infty \int_0^\infty e^{-f(x,y)} \frac{\partial^2 f(x,y)}{\partial x \, \partial y} \, \mathrm{d}x \, \mathrm{d}y,$$ where $f:\mathbb{R^{+}}\times\mathbb{R^{+}} \mapsto \mathbb{R^{+}}$ is differentiable infinitely many times. Moreover I know that for any $y \in [0,\infty)$, $\lim_{x\mapsto \infty} f(x,y) = 0$, for any $x \in [0,\infty)$, $\lim_{y\mapsto \infty} f(x,y) = 0$ and that the integral exists finite.
Do you have some technique to suggest to attack this kind of integral or similar bivariate integrals?
Thanks a lot!