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) \[2l=\int_{0}^{1}{\log \left\{ \frac{1-x}{x}\cdot \frac{x}{1-x} \right\}dx=}\int_{0}^{1}{\log 1\,dx=0}\] \[\Rightarrow \] \[l=0\]
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