Mathematics

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


SOLUTION
$$Let,\quad I=\int { \dfrac {\ dx }{ { \left( 1+{ x }^{ 2 } \right)  }^{ 2 } }  } \\ we\quad will\quad use\quad the\quad substitution\quad x=\tan \theta \\ \Rightarrow \ dx={ \sec }^ { 2 }\theta d\theta \\  =\int { \dfrac { { \sec }^ { 2 }\theta d\theta  }{ { \sec }^ { 4 }\theta  }  } \quad \quad \left[ \because 1+{ \tan }^ { 2 }\theta ={ \sec }^ { 2 }\theta  \right] \\ =\int { \dfrac { d\theta  }{ { \sec }^ { 2 }\theta  }  } \\ =\int { { \cos }^ { 2 }\theta d\theta  } \\ =\dfrac { 1 }{ 2 } \int { \cos 2\theta d\theta +\int { \dfrac { 1 }{ 2 } d\theta  }  } \quad \quad \left[ \because \cos 2\theta =2{ \cos }^ { 2 }\theta -1\Rightarrow { \cos }^ { 2 }\theta =\dfrac { 1 }{ 2 } \cos 2\theta +\dfrac { 1 }{ 2 }  \right] \\ =\dfrac { 1 }{ 4 } \sin 2\theta +\dfrac { 1 }{ 2 } \theta +C\\$$ 
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Subjective Medium Published on 17th 09, 2020
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