Question

$\underset{-1}{\overset{1}{\int }}\log \left(\frac{2-x}{2+x}\right)dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_{-1}^1\log \frac{2-x}{2+x}dx \\ \mathrm{Here}f\left(x\right)=\log \left(\frac{2-x}{2+x}\right) \\ f\left(-x\right)=\log \left(\frac{2+x}{2-x}\right) \\ =-\log \left(\frac{2-x}{2+x}\right) \\ =-f\left(x\right) \\ \mathrm{Hence}f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}, \\ \mathrm{Therefore}, \\ I=0 \end{gathered}$$