\begin{align} \int\int{\exp(-k|\vec x_1-\vec x_2|)d^2\vec x_1d^2\vec x_2} \end{align} where $k>0$ of course.
The notation $d^n q_k$ to indicate integral over the $n$ coordinates of a point in $\mathbb{R}^n$.
I tried using polar coordinates transformation , \begin{align} \int d^2 x= \int_0^\infty \!\! x ~dx ~\int _0^{2\pi} d\theta \end{align} But this does not simplify the integral.
How to perform the integration of the integral above by changing the variable to the center of mass and relative coordinate,as below, \begin{align} u&=(x_2+x_1)\\ v&=\frac{1}{2}(x_2-x_1) \end{align}