Question Number 48177 by Abdo msup. last updated on 20/Nov/18
$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\int_{{x}+\mathrm{1}} ^{\mathrm{2}{x}+\mathrm{1}} \:\:\:\frac{{tarctan}\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{\mathrm{1}+\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$
Commented by kaivan.ahmadi last updated on 21/Nov/18
$$\mathrm{it}\:\mathrm{is}\:\mathrm{zero} \\ $$
Commented by Abdo msup. last updated on 21/Nov/18
$${let}\:{A}\left({x}\right)=\int_{{x}+\mathrm{1}} ^{\mathrm{2}{x}+\mathrm{1}} \:\:\frac{{t}}{\mathrm{1}+\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{arctan}\left({t}^{\mathrm{2}} +\mathrm{1}\right){dt}\:{by}\:{parts} \\ $$$${u}^{'} =\frac{{t}}{\mathrm{1}+\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{and}\:{v}={arctan}\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)\Rightarrow \\ $$$${A}\left({x}\right)=\left[\frac{\mathrm{1}}{\mathrm{2}}{arctan}\left(\mathrm{1}+{t}^{\mathrm{2}} \right).{arctan}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right]_{{x}+\mathrm{1}} ^{\mathrm{2}{x}+\mathrm{1}} \\ $$$$−\int_{{x}+\mathrm{1}} ^{\mathrm{2}{x}+\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{2}}{arctan}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\:\frac{\mathrm{2}{t}}{\mathrm{1}+\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\:{arctan}^{\mathrm{2}} \left(\mathrm{1}+\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} \right)−{arctan}\left(\mathrm{1}+\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)\right) \\ $$$$−{A}\left({x}\right)\:\Rightarrow \\ $$$${A}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left\{\:{arctan}^{\mathrm{2}} \left(\mathrm{1}+\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} \right)−{arctan}\left(\mathrm{1}+\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)\right\}\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} {A}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left\{{arctan}^{\mathrm{2}} \left(\mathrm{2}\right)−{arctan}^{\mathrm{2}} \left(\mathrm{2}\right)\right\}\:=\mathrm{0}. \\ $$