Mathematics

$$\displaystyle \int_0^{\pi / 2} \dfrac{1}{1 + \sqrt [  4]{tanx  }}dx = $$ 


ANSWER

$$\pi / 4$$


SOLUTION
$$I=\int _{ 0 }^{ \pi  }{ \cfrac { 1 }{ 1+{ \left( \tan { x }  \right)  }^{ 1/4 } }  } dx=\int _{ 0 }^{ \cfrac { \pi  }{ 2 }  }{ \cfrac { 1 }{ 1+{ \left( \tan { x }  \right)  }^{ 1/4 } }  } dx\quad $$
$$2I=\int _{ 0 }^{ \cfrac { \pi  }{ 2 }  }{ \left( \cfrac { 1 }{ 1+{ \left( \tan { x }  \right)  }^{ 1/4 } } +\cfrac { { \left( \tan { x }  \right)  }^{ 1/4 } }{ 1+{ \left( \tan { x }  \right)  }^{ 1/4 } }  \right)  } dx=\int _{ 0 }^{ \cfrac { \pi  }{ 2 }  }{ \left( \cfrac { 1+{ \left( \tan { x }  \right)  }^{ 1/4 } }{ 1+{ \left( \tan { x }  \right)  }^{ 1/4 } }  \right)  } dx$$
$$2I=\int _{ 0 }^{ \cfrac { \pi  }{ 2 }  }{ 1 } dx\Rightarrow 2I=\cfrac { \pi  }{ 2 } $$
$$\Rightarrow I=\cfrac { \pi  }{ 4 } $$
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Single Correct Medium Published on 17th 09, 2020
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