What's the proof that $\int_0^\infty \tan x\,dx=\ln 2$?

Question

What's the proof that $\int_0^\infty \tan x\,dx=\ln 2$?

https://mathoverflow.net/a/349627/10059

Answer 1

The $0$-at-$0$ antiderivative $\ln|\sec x|$ is periodic with mean $\frac{1}{\pi}\int_0^\pi\ln\sec xdx$, which is famously $\ln 2$. If we regard the antiderivative as equal to its own mean at infinity, $\int_0^\infty\tan xdx=\ln 2$.