$I = \int\limits_0^1 {\frac{{dx}}{{{{\left( {\sqrt[5]{{1 + {x^5}}}} \right)}^6}}}} = \int\limits_0^1 {\frac{{dx}}{{\sqrt[5]{{1 + {x^5}}}}}} - \int\limits_0^1 {\frac{{{x^5}dx}}{{{{\left( {\sqrt[5]{{1 + {x^5}}}} \right)}^6}}}} $Ta tính $K = \int\limits_0^1 {\frac{{dx}}{{\sqrt[5]{{1 + {x^5}}}}}} $
Đặt: \[u = \frac{1}{{\sqrt[5]{{1 + {x^5}}}}} \Rightarrow du = - \frac{{{x^4}}}{{{{\left( {\sqrt[5]{{1 + {x^5}}}} \right)}^6}}}dx\]
\[dv = dx \Leftarrow v = x\]
$ \Rightarrow K = \frac{x}{{\sqrt[5]{{1 + {x^5}}}}}_0^1 + \int\limits_0^1 {\frac{{{x^5}dx}}{{{{\left( {\sqrt[5]{{1 + {x^5}}}} \right)}^6}}}} = \frac{1}{{\sqrt[5]{2}}} + \int\limits_0^1 {\frac{{{x^5}dx}}{{{{\left( {\sqrt[5]{{1 + {x^5}}}} \right)}^6}}}} $
$ \Rightarrow I = \frac{1}{{\sqrt[5]{2}}}$