Mathematics

$$\displaystyle\int \dfrac{1}{1+\sin^2 x}dx$$


SOLUTION
$$\int { \cfrac { 1 }{ \sin ^{ 2 }{ x }  }  } dx=\int { \cfrac { \sec ^{ 2 }{ x }  }{ \left( \cfrac { 1 }{ \cos ^{ 2 }{ x }  } +\tan ^{ 2 }{ x }  \right)  }  } dx=\int { \cfrac { \sec ^{ 2 }{ x }  }{ \sec ^{ 2 }{ x } +\tan ^{ 2 }{ x }  }  } dx\quad $$
put $$t=\tan { x } \Rightarrow dt=\sec ^{ 2 }{ x } dx$$
$$\Rightarrow \int { \cfrac { 1 }{ 1+2{ t }^{ 2 } }  } dx=\int { \cfrac { 1 }{ 1+{ \left( \sqrt { 2 } t \right)  }^{ 2 } }  } dt\Rightarrow \cfrac { 1 }{ \sqrt { 2 }  } \tan ^{ -1 }{ \sqrt { 2 }  } t\Rightarrow \cfrac { 1 }{ \sqrt { 2 }  } \tan ^{ -1 }{ \left( \sqrt { 2 } \tan { x }  \right)  } $$
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Subjective Medium Published on 17th 09, 2020
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