You can obtain separate "principal values" of the integral on the right and left sides by regularizing the improper integral. They, in general, will not coincide.

There are multiple methods of regularization.

Then you can find the average between the two obtained values.

This formula is of some help in this respect:
$$\int_0^\infty f(x) dx=\int_0^\infty \frac1{x^2}f\left(\frac1x\right)dx$$
You can convert the divergent integral near the pole into a divergent integral over infinite range, which can be compared or decomposed to divergent series (for instance, using Euler-Maclaurin formula). The divergent series then can be regularized using Borel or Ramanujan.