Mathematics

$$\int_0^{\pi/2} \dfrac{tan^7 x}{cot^7 x + tan^7 x} dx$$ is equal to


ANSWER

$$\dfrac{\pi}{4}$$


SOLUTION
$$I=\int _{ 0 }^{ { \pi  }/{ 2 } }{ \cfrac { \tan ^{ 7 }{ x }  }{ \cot ^{ 7 }{ x } +\tan ^{ 7 }{ x }  }  } dx\longrightarrow 1$$
$$\int _{ a }^{ b }{ f\left( x \right)  } dx=\int _{ a }^{ b }{ f\left( a+b-x \right)  } dx$$
$$\therefore I=\int _{ 0 }^{ { \pi  }/{ 2 } }{ \cfrac { \tan ^{ 7 }{ \left( \cfrac { \pi  }{ 2 } -x \right)  }  }{ \cot ^{ 7 }{ \left( \cfrac { \pi  }{ 2 } -x \right)  } +\tan ^{ 7 }{ \left( \cfrac { \pi  }{ 2 } -x \right)  }  }  } $$
$$I=\int _{ 0 }^{ { \pi  }/{ 2 } }{ \cfrac { \cot ^{ 7 }{ x }  }{ \tan ^{ 7 }{ x } +\cot ^{ 7 }{ x }  }  } dx\longrightarrow 2$$
Add equations 1 and 2
$$\Longrightarrow 2I=\int _{ 0 }^{ { \pi  }/{ 2 } }{ \left( \cfrac { \tan ^{ 7 }{ x }  }{ \tan ^{ 7 }{ x } +\cot ^{ 7 }{ x }  } +\cfrac { \cot ^{ 7 }{ x }  }{ \tan ^{ 7 }{ x } +\cot ^{ 7 }{ x }  }  \right)  } $$
$$\Longrightarrow 2I=\int _{ 0 }^{ { \pi  }/{ 2 } }{ 1 } \cdot dx$$
$$\Longrightarrow 2I={ \left| x \right|  }_{ 0 }^{ { \pi  }/{ 2 } }$$
$$\Longrightarrow 2I=\cfrac { \pi  }{ 2 } $$
$$I=\cfrac { \pi  }{ 4 } $$
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Single Correct Medium Published on 17th 09, 2020
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