How to show that $\int_{-\infty}^{\infty} \mathrm{d}^3 \textbf{k} \frac {e^{i \textbf{k x}}} {(2 \pi)^3} = \delta^3(x)$ in spherical coordinates?

Question

Recently I had to deal with Fourier transformations and delta functions, and I was wondering how about that. I know, that its trivial to show in cartesian coordinates, but i couldn't do it in spherical coordinates. Somehow one should be able to reduce it to something which only depends on |k| (which i could do) and show that it is a delta function (which i couldn't).