Question Number 36426 by abdo.msup.com last updated on 01/Jun/18
$${find}\:\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{1}+{x}^{\mathrm{3}} \:}\:{dx} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 02/Jun/18
$$\int{x}^{\mathrm{2}} \sqrt{\mathrm{1}+{x}^{\mathrm{3}} \:}\:\:{dx} \\ $$$${t}=\mathrm{1}+{x}^{\mathrm{3}} \\ $$$${dt}=\mathrm{3}{x}^{\mathrm{2}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\int\sqrt{{t}}\:{dt} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}×\frac{{t}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$$\frac{\mathrm{2}}{\mathrm{9}}\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$$ \\ $$