Cauchy principal value and the "normal" definition.

Question

Suppose that $\int^{\infty}_{-\infty}f(x)\, dx$ exist. How to prove that $\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$ also exist, and $\int^{\infty}_{-\infty}f(x)\, dx=\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$

Answer 1

$\int_{-\infty}^{\infty}f(x)\, dx$ exist mean that both $$ \int_{-\infty}^{0}f(x)\, dx,\,\int_{0}^{\infty}f(x)\, dx $$

exist

This means that the two limits

$$ \lim_{b\to\infty}\int_{-b}^{0}\, f(x)\, dx,\,\lim_{b\to\infty}\int_{0}^{b}\, f(x)\, dx $$

exist.

Since the two limits exists you can add them up and get the desired result