Question Number 205294 by depressiveshrek last updated on 15/Mar/24
$$\int\sqrt[{\mathrm{3}}]{\mathrm{3}{x}−{x}^{\mathrm{3}} }{dx} \\ $$
Answered by Frix last updated on 15/Mar/24
$$\int\left(\mathrm{3}{x}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}\:\overset{{t}=\frac{{x}^{\mathrm{2}} }{\mathrm{3}}} {=}\:\frac{\mathrm{3}}{\mathrm{2}}\int{t}^{−\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}−{t}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dt}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}{t}^{\frac{\mathrm{2}}{\mathrm{3}}} \:_{\mathrm{2}} {F}_{\mathrm{1}} \:\left(−\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{2}}{\mathrm{3}};\:\frac{\mathrm{5}}{\mathrm{3}};\:{t}\right)\:= \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\mathrm{3}{x}^{\mathrm{4}} }}{\mathrm{2}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \:\left(−\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{2}}{\mathrm{3}};\:\frac{\mathrm{5}}{\mathrm{3}};\:\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\right)\:+{C} \\ $$