Question Number 146689 by naka3546 last updated on 15/Jul/21
$$\int\:\:\frac{−\mathrm{3}{x}+\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{10}{x}\right)^{\mathrm{2}} }}\:\:{dx}\:\:=\:\:\:...\:\:? \\ $$$${Solve}\:\:{it}\:\:{without}\:\:{substitution}\:\:{method}. \\ $$
Answered by Ar Brandon last updated on 15/Jul/21
$$\mathcal{I}=\int\frac{−\mathrm{3x}+\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3x}^{\mathrm{2}} −\mathrm{10x}\right)^{\mathrm{2}} }}\mathrm{dx} \\ $$$$\:\:\:=−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{6x}−\mathrm{10}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3x}^{\mathrm{2}} −\mathrm{10x}\right)^{\mathrm{2}} }}\mathrm{dx} \\ $$$$\:\:\:=−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{d}\left(\mathrm{3x}^{\mathrm{2}} −\mathrm{10x}\right)}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3x}^{\mathrm{2}} −\mathrm{10x}\right)^{\mathrm{2}} }} \\ $$$$\:\:\:=−\frac{\mathrm{3}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{3x}^{\mathrm{2}} −\mathrm{10x}}+\mathrm{C} \\ $$
Commented by naka3546 last updated on 15/Jul/21
$${thank}\:{you},\:{sir}. \\ $$