I am having trouble formulating an example for which
$\mathcal{P}\int^{\infty}_{-\infty}f(x)dx\neq\int^{\infty}_{-\infty}f(x)dx$
Would an example be $f(x)=1/x$ because of the asymptote at $x=0$?
2026-04-04 05:19:02.1775279942
Cauchy Principal Value different from the improper integral
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Yes, $1/x$ (restrict to $[-1,1]$ is you wish) is odd so its Cauchy Principal Value is 0, but its not integrable so its integral is not well defined.