Question

$\underset{0}{\overset{1}{\int }}\log (\frac{1}{x}-1)dx=$

• A. $1$
• B. $0$
• C. $2$
• D. None of these

Answer 1

The correct option is B $0$

$I=\underset{0}{\overset{1}{\int }}\log (\frac{1}{x}-1)dx=\underset{0}{\overset{1}{\int }}\log \left(\frac{1-x}{x}\right)dx=\underset{0}{\overset{1}{\int }}\log \left(\frac{x}{1-x}\right)dx=-I\cdots \left(\text{using}\underset{a}{\overset{b}{\int }}f\right(x)dx=\underset{a}{\overset{b}{\int }}f(a+b-x\left)dx\right)\Rightarrow 2I=0\Rightarrow 2I=0$