Integration over wave number

Question

I came across an integral of this form in a paper: $$ g_j\int\frac{m_{jp}(k,\omega)}{-i\omega+k^2}d\mathbf{k}d\omega, $$ where the second-rank tensor $m_{jp}(k,\omega)$ is proportional to: $$ m_{jp}(k,\omega)\propto\delta_{jp}-k_jk_p/k^2. $$ The result should be an integral of the form: $$ \mathbf{g}\int_0^{\infty}f(k,\omega)dkd\omega. $$ The whole derivation is rather complicated and there aren't a lot of explanations. I can't figure out how the integral over $\mathbf{k}$ is turned into one for $k$ and the $\delta_{jp}$ and $k_j$, $k_p$ terms are delt with. The function $f$ is a complicated function of logarithms and arctangents, so I assume the step involves quite a bit of math.