$\int\limits_{0}^{1}{\frac{d\text{x}}{\sqrt[5]{{{(1+{{x}^{5}})}^{6}}}}=\int\limits_{0}^{1}{\frac{(1+{{x}^{5}})-{{x}^{5}}}{(1+{{x}^{5}})\sqrt[5]{1+{{x}^{5}}}}d\text{x}=\int\limits_{0}^{1}{\frac{d\text{x}}{\sqrt[5]{1+{{x}^{5}}}}-\int\limits_{0}^{1}{x(\frac{{{x}^{4}}d\text{x}}{ \sqrt[5]{{{(1+{{x}^{5}})}^{6}}}})d\text{x}}}}}$
$\left\{\begin{matrix} u=x
& \\ dv=\frac{{{x}^{4}}}{\sqrt[5]{{{(1+{{x}^{5}})}^{6}}}}dx
&
\end{matrix}\right.$
$\Rightarrow \left\{\begin{matrix} du=dx
& \\ v=\int{\frac{{{x}^{4}}}{\sqrt[5]{{{(1+{{x}^{5}})}^{6}}}}dx=\frac{-1}{\sqrt[5]{1+{{x}^{5}}}}}
&
\end{matrix}\right.$
$\int\limits_{0}^{1}{\frac{{{x}^{5}}d\text{x}}{ \sqrt[5]{{{(1+{{x}^{5}})}^{6}}}}}=\left. \frac{-x}{\sqrt[5]{1+{{x}^{5}}}} \right|+\int\limits_{0}^{1}{\frac{d\text{x}}{\sqrt[5]{1+{{x}^{5}}}}=-\frac{1}{\sqrt[5]{2}}+{{I}_{2}}}$
$I=\frac{1}{\sqrt[5]{2}}$
Suy ra
$I={{I}_{2}}-{{I}_{2}}-(-\frac{1}{\sqrt[5]{2}})=\frac{1}{\sqrt[5]{2}}$
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