Find the limit of this integral:

Solution 1:

If $t>1$, then $\arctan(t) \in (\pi/4,\pi/2)$ and hence $$ \frac{2}{\pi}<\frac{1}{\arctan(t)}<\frac{4}{\pi} $$ and $$ \frac{2\cdot 2x}{\pi}<\int_x^{3x}\frac{dt}{\arctan(t)}<\frac{4\cdot 2x}{\pi} $$ Thus $$ \lim_{x\to\infty}\int_x^{3x}\frac{dt}{\arctan(t)}=\infty. $$

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