As far as I know, the principal value of a non-summable function like $1/x$, denoted $\mathcal{P}(1/x)$, is a distribution that that acts on some smooth function $f$ in some test-function space and gives back the number:
\begin{equation} \mathcal{P}(1/x)\,\,f=\lim_{\epsilon\to 0}\int_{\mathcal{R}-[-\epsilon,\epsilon]}\frac{f(x)}{x}\,dx \end{equation}
How would you prove rigorously that this is equivalent to:
\begin{equation} \mathcal{P}(1/x)\,\,f=\lim_{\epsilon\to 0}\int\frac{x\,f(x)}{x^2+\epsilon^2}\,dx \end{equation}