Cauchy Principal Value in R

The Cauchy Principal value is a is a method for assigning values to certain improper integrals which would otherwise be undefined.

For a singularity at a finite number b

We can also define over singularities. Let $a<b<c$, where $b$ is a difficult point at which the behaviour of the $f$ is such that

for any $a<b$, and

Then we can define the Cauchy Principle value for a singularity at a finite number $b$, as

For a singularity at $\mathrm{\infty }$

The Cauchy principle Value at infinity is defined as:

where

and

In some cases it is necessary to deal simultaneously with singularities both at a finite number $b$ and at infinity. This is usually done by a limit of the form

In those cases where the integral may be split into two independent, finite limits,

and

According to the my Complex Analysis by Paez:
Given a function $f:\mathbb{R}\{x_1,\dots ,x_n\}\to \mathbb{R}$ such that $x_k<x_{k+1}$ for $k\in \{1,\dots ,n-1\}$ and $f$ is not bounded for any neighbourhood around $x_k$, and integrable over any interval $[c,d]\subseteq \mathbb{R}\setminus \{x_1,\dots ,x_n\}$, we define the Cauchy principal value of the integral ${\int }_{\mathbb{R}}f$ as

which can be the same from the definition from wikipedia but don't know how.

Calculating the Cauchy principal value can be useful, since if $f$ is even and integrable over any interval in $\mathbb{R}$ we get that