Question Number 227271 by Spillover last updated on 11/Jan/26
Answered by Kassista last updated on 11/Jan/26
$$ \\ $$$$\int\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}−\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:−\int\frac{\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\: \\ $$$$\overset{{x}=\sqrt{\mathrm{5}}\mathrm{tan}\:\theta} {\Rightarrow}\:{x}\:−\mathrm{5}\:\int\:\frac{\sqrt{\mathrm{5}}\mathrm{sec}\:^{\mathrm{2}} \theta}{\:\mathrm{5sec}^{\mathrm{2}} \:\theta}\:{d}\theta\:=\:{x}−\sqrt{\mathrm{5}}\int\:{d}\theta\: \\ $$$$=\:{x}\:−\sqrt{\mathrm{5}}{arctan}\left(\frac{{x}}{\:\sqrt{\mathrm{5}}}\right)+{C} \\ $$
Answered by Spillover last updated on 11/Jan/26
Answered by Spillover last updated on 11/Jan/26
