(The Cauchy principal value of) $$ \int_0^{\infty}\frac{\tan x}{x}\mathrm dx $$
I tried to cut this integral into $$\sum_{k=0}^{\infty}\int_{k\pi}^{(k+1)\pi}\frac{\tan x}{x}\mathrm dx$$ And then $$\sum_{k=0}^{\infty}\lim_{\epsilon \to 0}\int_{k\pi}^{(k+1/2)\pi-\epsilon}\frac{\tan x}{x}\mathrm dx+\int_{(k+1/2)\pi+\epsilon}^{(k+1)\pi}\frac{\tan x}{x}\mathrm dx$$ $$\sum_{k=0}^{\infty}\int_{k\pi}^{(k+1/2)\pi}\frac{((2k+1)\pi-2x)\tan x}{((2k+1)\pi-x)x}\mathrm dx$$ And I did not know how to continue. I did not know if I was right or not. How to calculate this integral?