Mathematics

$$\displaystyle \int_{0}^{\infty }\frac{x}{\left ( 1+x \right )\left ( 1+x^{2} \right )}dx$$


ANSWER

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

is sme as $$\displaystyle \int_{0}^{\infty }\frac{dx}{\left ( 1+x \right )\left ( 1+x^{2} \right )}$$


SOLUTION
Let $$I=\int _{ 0 }^{ \infty  }{ \cfrac { xdx }{ \left( 1+x \right) \left( 1+{ x }^{ 2 } \right)  }  } $$
Using partial fraction
$$=\int _{ 0 }^{ \infty  }{ \left( \cfrac { x+1 }{ 2\left( 1+{ x }^{ 2 } \right)  } -\cfrac { 1 }{ 2\left( 1+x \right)  }  \right) dx } \\ ={ \left( \lim _{ b\rightarrow \infty  }{ \cfrac { 1 }{ 2 } \log { \left( 1+{ x }^{ 2 } \right)  } -\cfrac { 1 }{ 2 } \log { \left( 1+x \right)  } +\cfrac { 1 }{ 2 } { tan }^{ -1 }x }  \right)  }_{ 0 }^{ b }\\ =\lim _{ b\rightarrow \infty  }{ \left( \cfrac { 1 }{ 2 } \log { \left( 1+b \right)  } -\cfrac { 1 }{ 2 } \log { \left( 1+b \right)  } +\cfrac { 1 }{ 2 } { tan }^{ -1 }b \right)  } \\ =\cfrac { \pi  }{ 4 } $$
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Multiple Correct Medium Published on 17th 09, 2020
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