Question

Evaluate :$\underset{0}{\overset{\infty }{\int }}\log \left(x+\frac{1}{x}\right)\frac{dx}{1+x^2}$

Answer 1

$\begin{array}{c}{\int }_0^{\infty }\log \left(x+\frac{1}{x}\right)\frac{dx}{1+x^2}\\ Now,\\ x=\tan \theta dx={\sec }^2\theta d\theta \\ I={\int }_0^{\frac{\pi }{2}}\frac{\log \left(\tan \theta +\cot \theta \right)}{1+{\tan }^2\theta }{\sec }^2\theta d\theta \\ =-{\int }_0^{\frac{\pi }{2}}\log \sin \theta d\theta -{\int }_0^{\frac{\pi }{2}}\log \left(cos\theta \right)d\theta \\ =\pi \log 2\end{array}$