Question Number 112773 by 675480065 last updated on 09/Sep/20
$$\int\frac{\:\sqrt{{x}}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx} \\ $$$${Please}\:{help} \\ $$
Answered by MJS_new last updated on 09/Sep/20
$$\int\frac{\sqrt{{x}}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}={x}^{\mathrm{3}/\mathrm{2}} \:\rightarrow\:{dx}=\frac{\mathrm{2}{dt}}{\mathrm{3}\sqrt{{x}}}\right] \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\int\frac{{dt}}{{t}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{2}}{\mathrm{3}}\mathrm{arctan}\:{t}\:= \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\mathrm{arctan}\:{x}^{\mathrm{3}/\mathrm{2}} \:+{C} \\ $$