I am looking over some notes for a class and the professor left the following as a side note: $$\int \exp{\left( -\beta f(|\mathbf{x}_1-\mathbf{x}_2|)\right)}\; d^3\mathbf{x}_1 d^3\mathbf{x}_2 = V \int \exp{\left( -\beta f(r_{12})\right)} \;dr_{12}$$ Where $\mathbf{x}_1$ and $\mathbf{x}_2$ are both Cartesian points in some volume $V$ and $\beta$ is a constant.
I attempted to prove this myself, but I have had little luck. I thought about doing a change of variables as follows: $$r_{12} = |\mathbf{x}_1-\mathbf{x}_2| \implies\frac{dr_{12}}{d\mathbf{x}_1} = \frac{\mathbf{x}_1-\mathbf{x}_2}{r_{12}}\text{ and }\frac{dr_{12}}{d\mathbf{x}_2} = \frac{\mathbf{x}_2-\mathbf{x}_1}{r_{12}}$$ but im not sure what else you could do after that. Any ideas?