If $ P(1/x) $ means the principal value of $1/x$ and $ H(x) $ is the Heaviside step function is this then correct (regularization)
$$\operatorname{P.V.}\int_{-\infty}^{\infty}\frac{H(x)f(x)}{x}\mathrm dx=\int_{0}^{\infty}\frac{H(x)f(x)-H(0)f(0)}{x}\mathrm dx.$$