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Question Number 54821 by Abdo msup. last updated on 12/Feb/19

$$find{lim}_{n\to +\mathrm{\infty }}{\int }_0^n\frac{arctan\left(nx\right)}{n^2+x^2}dx$$

Commented by maxmathsup by imad last updated on 13/Feb/19

$$let{A}_n={\int }_0^n\frac{arctan\left(nx\right)}{n^2+x^2}dx\Rightarrow A_n{=}_{x=nt}{\int }_0^1\frac{arctan\left(n^{2}t\right)}{n^2(1+t^2)}ndt$$$$=\frac{1}{n}{\int }_0^1\frac{arctan\left(n^{2}t\right)}{1+t^2}dtbut{lim}_{n\to +\mathrm{\infty }}{\int }_0^1\frac{arctan\left(n^{2}t\right)}{1+t^2}dt=\frac{\pi }{2}{\int }_0^1\frac{dt}{1+t^2}$$$$=\frac{\pi }{2}\frac{\pi }{4}=\frac{{\pi }^2}{8}and{lim}_{n\to +\mathrm{\infty }}\frac{1}{n}=0\Rightarrow {lim}_{n\to +\mathrm{\infty }}{A}_n=0.$$

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