Question

$$
\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x=
$$
(A) 1
(B) 0
(C) 2
(D) None of these

Answer 1

Ans: (B)
 Let I \(=\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x=\int_{0}^{1} \log \left(\frac{1-x}{x}\right) d x\)
$$
\begin{aligned}
&I=\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x=-I \\
&\therefore 2 I=0 \Rightarrow I=0
\end{aligned}
$$