Mathematics

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


SOLUTION

Consider the given function:

  $$ f(x)=\int_{0}^{\infty }{\dfrac{\log (1+{{x}^{2}})}{(1+{{x}^{2}})}}dx $$

 $$ \,\,\,\,\,\,\,\,\,\,\,\,let\,\,x=\sec \theta  $$

 $$ =\int_{0}^{\dfrac{\pi }{2}}{{{\sec }^{2}}\theta \dfrac{\log ({{\sec }^{2}}\theta )}{{{\sec }^{2}}\theta }}\,\,d\theta  $$

 $$ =\int_{0}^{\dfrac{\pi }{2}}{2\log {{\sec }^{2}}\theta }\,\,d\theta  $$

 $$ =\int_{0}^{\dfrac{\pi }{2}}{-2\log \cos \theta }\,\,d\theta  $$

 $$ =-2*\int_{0}^{\dfrac{\pi }{2}}{\log \cos \theta \,\,d\theta } $$

 $$ =-2*(-\dfrac{\pi }{2})\log 2 $$

 $$ =\pi \log 2 $$

Hence this is the answer.

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Subjective Medium Published on 17th 09, 2020
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