Mathematics

Solve :$$\displaystyle \int \dfrac{\sqrt{1-x^3}}{x^2\sqrt{x}}dx.$$ ?


SOLUTION
$$\quad \int { \dfrac { \sqrt { 1-{ x }^{ 3 } }  }{ { x }^{ 2 }\sqrt { x }  } dx } \\ =\left( \dfrac { 1 }{ x\sqrt { x }  } -\dfrac { 5 }{ 3{ x }^{ \dfrac { 3 }{ 2 }  } }  \right) \sqrt { 1-{ x }^{ 3 } } -\int { \sqrt { \dfrac { x }{ 1-{ x }^{ 3 } }  }  } dx\quad \quad \quad \quad \left[ applying\quad integration\quad by\quad parts \right] \\ =\left( \dfrac { 1 }{ x\sqrt { x }  } -\dfrac { 5 }{ 3{ x }^{ \dfrac { 3 }{ 2 }  } }  \right) \sqrt { 1-{ x }^{ 3 } } -\dfrac { 2 }{ 3 } \sin^{-1} { \left( { x }^{ \dfrac { 3 }{ 2 }  } \right)  } +C$$
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Subjective Medium Published on 17th 09, 2020
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