Question Number 32269 by abdo imad last updated on 22/Mar/18
$${find}\:\int\:\:\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$
Answered by mrW2 last updated on 22/Mar/18
$$\int\:\:\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\:\:\frac{{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\:\:\frac{{x}^{\mathrm{2}} +\mathrm{1}−\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\int\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}^{\mathrm{2}} −\:\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\int\:\sqrt{{t}+\mathrm{1}}\:{dt}−\:\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{t}}}\:{dt}\right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{\mathrm{2}}{\mathrm{3}}\left({t}+\mathrm{1}\right)\sqrt{{t}+\mathrm{1}}−\mathrm{2}\sqrt{{t}+\mathrm{1}}\right\} \\ $$$$\left\{\frac{\mathrm{1}}{\mathrm{3}}\left({t}+\mathrm{1}\right)−\mathrm{1}\right\}\sqrt{{t}+\mathrm{1}} \\ $$$$\frac{\left({t}−\mathrm{2}\right)\sqrt{{t}+\mathrm{1}}}{\mathrm{3}}+{C} \\ $$$$\frac{\left({x}^{\mathrm{2}} −\mathrm{2}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{\mathrm{3}}+{C} \\ $$