$ \int_{-\infty}^{\infty} \frac{1}{x} \left( \frac{\exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right)}{\sqrt{2\pi}\sigma} \right) dx $

Question

I've encountered an intriguing problem involving integrals of functions derived from the probability density functions (PDFs) of normal distributions. Initially, I am considering the following integral:

$ \int_{-\infty}^{\infty} \frac{1}{x} \left( \frac{\exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right)}{\sqrt{2\pi}\sigma} \right) dx $

This integral can be expressed as:

$ \int_{0}^{\infty} \frac{1}{x} \left( \frac{\exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right)}{\sqrt{2\pi}\sigma} - \frac{\exp\left(-\frac{(-x - \mu)^2}{2\sigma^2}\right)}{\sqrt{2\pi}\sigma} \right) dx $

Which simplifies to:

$ \int_{0}^{\infty} \frac{1}{x} \left( \text{pdf}(x; \mu, \sigma^2) - \text{pdf}(-x; \mu, \sigma^2) \right) dx $

My question is about the generalizability and convergence of these integrals. Specifically, I am interested in understanding if these integrals can be generalized for any values of $\mu$ and $\sigma$, and under what conditions they converge.

Given the nature of these integrals, involving the normal distribution and a division by $x$, I am unsure how to approach their analysis, particularly at the lower bound where the function may potentially diverge.

Any insights or references to similar problems would be greatly appreciated.