2022-07-10

Consider the following:
${\int }_{x}^{3x}\frac{1}{\mathrm{arctan}t}dt$

potamanixv

Expert

If $t>1$, then $\mathrm{arctan}\left(t\right)\in \left(\pi /4,\pi /2\right)$ and hence
$\frac{2}{\pi }<\frac{1}{\mathrm{arctan}\left(t\right)}<\frac{4}{\pi }$
and
$\frac{2\cdot 2x}{\pi }<{\int }_{x}^{3x}\frac{dt}{\mathrm{arctan}\left(t\right)}<\frac{4\cdot 2x}{\pi }$
Thus
$\underset{x\to \mathrm{\infty }}{lim}{\int }_{x}^{3x}\frac{dt}{\mathrm{arctan}\left(t\right)}=\mathrm{\infty }.$

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