Given a probability density, how can I sample from the induced distribution?

Let f be an integratable function such that $\int f(x)dx=1$. If we want to take random samples from this, using whatever programming language one pleases, we should compute $F(t)={\int}_{-\mathrm{\infty}}^{t}f(x)dx$, invert this function and feed it numbers drawn from a uniform distribution on [0,1].

However I now want to sample coordinates $(u,v)\in {\mathbb{R}}^{2}\mathrm{\setminus}\{0\}$ such that the probability of (u,v) lying in a set E is given by

${\int}_{E}({u}^{2}+{v}^{2}{)}^{-\frac{3}{2}}dudv$

I don't see how I can now try to find

$F(s,t)={\int}_{-\mathrm{\infty}}^{s}{\int}_{-\mathrm{\infty}}^{t}({u}^{2}+{v}^{2}{)}^{-\frac{3}{2}}dudv$

as I get into troubles close to the zero. What other way is there to obtain a sample following this distribution?

A note for the context: If we consider all lines in the Euclidean plane with Cartesian coordinates, not passing through the origin, they can be represented via $ux+vx=1$. If we impose the condition that the probability densitiy should be invariant under Euclidean transformations, then we arrive at the above distribution.