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
![∫((√x)/(x^3 +1))dx= [t=x^(3/2) → dx=((2dt)/(3(√x)))] =(2/3)∫(dt/(t^2 +1))=(2/3)arctan t = =(2/3)arctan x^(3/2) +C](Q112799.png)
$$\int\frac{\sqrt{{x}}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}={x}^{\mathrm{3}/\mathrm{2}} \:\rightarrow\:{dx}=\frac{\mathrm{2}{dt}}{\mathrm{3}\sqrt{{x}}}\right] \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\int\frac{{dt}}{{t}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{2}}{\mathrm{3}}\mathrm{arctan}\:{t}\:= \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\mathrm{arctan}\:{x}^{\mathrm{3}/\mathrm{2}} \:+{C} \\ $$