Question Number 159477 by HongKing last updated on 17/Nov/21
$$\mathrm{Find}:\:\:\:\Omega\:=\int\:\frac{\mathrm{1}}{\left(\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$
Answered by MJS_new last updated on 17/Nov/21
$${t}=\mathrm{arctan}\:{x} \\ $$$$\Rightarrow \\ $$$$\int\mathrm{sin}^{\mathrm{2}} \:{t}\:{dt}=\frac{\mathrm{1}}{\mathrm{2}}\int\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2}{t}\right){dt}= \\ $$$$=\frac{{t}}{\mathrm{2}}−\frac{\mathrm{sin}\:\mathrm{2}{t}}{\mathrm{4}}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{arctan}\:{x}\:−\frac{{x}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}+{C} \\ $$
Commented by HongKing last updated on 17/Nov/21
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Ser} \\ $$