I have an integral of the form
$\int\nolimits^\infty_{-\infty}\mathrm d \omega \, \frac{\omega^2}{k^2 + \gamma^2 \omega^2}$
which diverges. This integral should have a finite value, as it must related to some physical measurement. I am trying to assign a value to the integral, kind of like how one does using regularisation. In a few papers on theoretical physics (Which is the field I am in), I have seen people use the Cauchy principal value in the form
$-\!\!\!\!\!\!\!\int^\infty_{-\infty}\mathrm dx \, f(x) = \lim_{L \to \infty} \, \frac1{L} \int^L_{-L}\mathrm dx \, f(x)$
but I am not sure how one deduces that from