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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
$\int_0^1 \log \left(\frac{1}{x}-1\right) d x$ is equal to :
  • ✓
    $0$
  • B
    1
  • C
    $\log 2$
  • D
    $\log \frac{3}{2}$
✓

Answer

Correct option: A.
$0$
(A)Let
$
\begin{aligned}
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 \ldots(1) \\
I & =\int_0^1 \log \left(\frac{1-x}{x}\right) d x \\
& =\int_0^1 \log \left(\frac{1-(1-x)}{(1-x)}\right) d x \text { From Property } P_5
\end{aligned}
$
$
\begin{aligned}
I & =\int_0^1 \log \left(\frac{1-1+x}{(1-x)}\right) d x=\int_0^1 \log \left(\frac{x}{(1-x)}\right) d x \\
I & =-\int_0^1 \log \left(\frac{1-x}{x}\right) d x \\
I & =-I \\
2 I & =0 \therefore I=0
\end{aligned}
$
Hence option (A) is correct.

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