Question

Evaluate $\underset{-1/2}{\overset{1/2}{\int }}\cos x\ln \left(\frac{1+x}{1-x}\right)dx$

Answer 1

Let, $f\left(x\right)=\cos \left(x\right)\cdot \ln \left(\frac{1+x}{1-x}\right)f(-x)=\cos (-x)\cdot \ln \left(\frac{1-x}{1+x}\right)=-\cos \left(x\right)\cdot \ln \left(\frac{1+x}{1-x}\right)=-f\left(x\right)$
$\Rightarrow f\left(x\right)$ is an odd function.
Hence, the value of the given intergral is $0$