Mathematics

Find the value of $$\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { x }  }  }{ \sqrt { \sin { x }  } +\sqrt { \cos { x }  }  } dx } $$


ANSWER

$$\displaystyle \frac { \pi  }{ 4 } $$


SOLUTION
$$\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { x }  }  }{ \sqrt { \sin { x }  } +\sqrt { \cos { x }  }  } dx } =\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { \left( \frac { \pi  }{ 2 } -x \right)  }  }  }{ \sqrt { \sin { \left( \frac { \pi  }{ 2 } -x \right)  }  } +\sqrt { \cos { \left( \frac { \pi  }{ 2 } -x \right)  }  }  } dx } $$   ....(1)

Using $$\displaystyle \int _{ a }^{ b }{ f\left( x \right)dx } =\int _{ a }^{ b }{ f\left( a+b-x \right)dx } $$

$$\displaystyle \Rightarrow I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \cos { x }  }  }{ \sqrt { \cos { x }  } +\sqrt { \sin { x }  }  } dx } $$   ...(2)

Adding (1) and (2)
$$\displaystyle 2I=\int _{ 0 }^{ \pi /2 }{ \left( \frac { \sqrt { \sin { x }  }  }{ \sqrt { \sin { x }  } +\sqrt { \cos { x }  }  } +\frac { \sqrt { \cos { x }  }  }{ \sqrt { \cos { x }  } +\sqrt { \sin { x }  }  }  \right)  } dx$$

$$\displaystyle =\int _{ 0 }^{ \pi /2 }{ \left( \frac { \sqrt { \sin { x }  } +\sqrt { \cos { x }  }  }{ \sqrt { \sin { x }  } +\sqrt { \cos { x }  }  }  \right) dx } =\int _{ 0 }^{ \pi /2 }{ dx } ={ \left[ x \right]  }_{ 0 }^{ \pi /2 }=\frac { \pi  }{ 2 } $$

$$\displaystyle \therefore I=\frac { \pi  }{ 4 } $$


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Single Correct Medium Published on 17th 09, 2020
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