= $\int\limits_{\frac{1}{\sqrt[3]{3}}}^{1}\frac{(x^{2}\times x )dx}{x^{3}\times (x^{3}+\frac{1}{3})\times \sqrt[3]{x^{3}+\frac{1}{3}}}$
= $\int\limits_{\frac{1}{\sqrt[3]{3}}}^{1}\frac{x^{2}dx}{x^{3}\times (x^{3}+\frac{1}{3})\times \sqrt{x^{3}+\frac{1}{3}}}$
Đặt $x=\frac{1}{t}$ $\Rightarrow dx=\frac{-dt}{x^{2}}$
Khi $x=\frac{1}{\sqrt[3]{3}} \Rightarrow t=\sqrt[3]{3}$
Khi $x=1 \Rightarrow t=1$
$\Rightarrow I=\int\limits_{\sqrt[3]{3}}^{1}\frac{\frac{-dt}{t^{2}}}{(\frac{1}{t^{3}}+\frac{1}{3})\times \sqrt[3]{\frac{1}{t^{3}}+\frac{1}{3}}} $
=$\int\limits_{1}^{\sqrt[3]{3}}\frac{\frac{dt}{t^{2}}}{\frac{3+t^{3}}{3t^{3}}\times \frac{\sqrt[3]{3+t^{3}}}{\sqrt[3]{3}t}}$
=$3\sqrt[3]{3}\times \int\limits_{1}^{\sqrt[3]{3}}\frac{t^{2}dt}{(3+t^{3})\times \sqrt[3]{3+t^{3}}}$
Đặt $u=\sqrt[3]{3+t^{3}} \Rightarrow u^{3}=t^{3}+3\Rightarrow u^{2}du=t^{2}dt$
Khi $t=1\Rightarrow u=\sqrt[3]{4}$
Khi $t=\sqrt[3]{3}\Rightarrow u=\sqrt[3]{6}$
$\Rightarrow I=3\sqrt[3]{3}\int\limits_{\sqrt[3]{4}}^{\sqrt[3]{6}}\frac{u^{2}du}{u^{4}}$
$=3\sqrt[3]{3}\times (\frac{1}{\sqrt[3]{4}}-\frac{1}{\sqrt[3]{6}})$