Mathematics

Evaluate: $$\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x }  }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x }  }  } dx$$


SOLUTION
$$I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x }  }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x }  }  } dx=\int_{3}^{9}{\cfrac{\sqrt[3]{12-(9+3-x)}}{\sqrt[3]{9+3-x}+\sqrt[3]{12-(9+3-x)}}}dx$$

$$I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x }  }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x }  }  } dx=\int_{3}^{9}{\cfrac{\sqrt[3]{x}}{\sqrt[3]{12-x}+\sqrt[3]{x}}}dx$$

$$2I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x }  }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x }  }  } dx+\int_{3}^{9}{\cfrac{\sqrt[3]{x}}{\sqrt[3]{12-x}+\sqrt[3]{x}}}dx$$

$$2I=\displaystyle\int_{3}^{9}dx=6\implies I=3$$

$$\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x }  }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x }  }  } dx=3$$
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Subjective Medium Published on 17th 09, 2020
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