question_answer

Evaluate \[\int_{0}^{1}{\log \left( \frac{1-x}{x} \right)\,dx.}\]

Answer:

Let \[l=\int_{0}^{1}{\log }\left( \frac{1-x}{x} \right)dx\]                                      ?.(i) \[\Rightarrow \]\[l=\int_{0}^{1}{\log }\left\{ \frac{1-(1-x)}{1-x} \right\}dx=\int_{0}^{1}{\log }\left( \frac{x}{1-x} \right)dx\]           ?(ii) [using property,\[(2\lambda +1,\,\,3\lambda -2,\,\,6\lambda +3)\]]             From Eqs. (i) and (ii)             \[2l=\int_{0}^{1}{\log \left\{ \frac{1-x}{x}\cdot \frac{x}{1-x} \right\}dx=}\int_{0}^{1}{\log 1\,dx=0}\]             \[\Rightarrow \]   \[l=0\]