Question Number 192406 by josemate19 last updated on 17/May/23
$$\int\:\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}{{x}}\:{dx}??? \\ $$
Answered by mehdee42 last updated on 17/May/23
$${x}=\mathrm{2}{chu}\Rightarrow{dx}=\mathrm{2}{shudu} \\ $$$$\Rightarrow{I}=\mathrm{2}\int\:\frac{{sh}^{\mathrm{2}} {u}}{{chu}}\:{du}\:=\mathrm{2}\int\:\left({chu}−\frac{\mathrm{1}}{{chu}}\:\right){du} \\ $$$$=\mathrm{2}\left({shu}−\mathrm{2}{tag}^{−\mathrm{1}} {e}^{{u}} \:\right)=\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}−\mathrm{4}{tag}^{−\mathrm{1}} \left(\frac{{x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}{\mathrm{2}}\:\right)\:+{c}\:\checkmark \\ $$