_ 1/e ^e | x| ,dx = - Vidyadip

MCQ

$\int_{1/e}^e {|\log x|\,dx = } $

A
$1 - \frac{1}{e}$
✓
$2\,\left( {1 - \frac{1}{e}} \right)$
C
${e^{ - 1}} - 1$
D
None of these

✓

Answer

Correct option: B.

$2\,\left( {1 - \frac{1}{e}} \right)$

b
(b) $\int_{1/e}^e {|\log x|dx = \int_{1/e}^1 { - \log x\,dx + \int_1^e {\,\log x\,dx} } } $

$ = [x - x\log x]_{1/e}^1 + [x\log x - x]_1^e$

$ = (1 - 0) - \left\{ {\frac{1}{e} - \frac{1}{e}( - 1)} \right\} + e - e + 1$

$ = 2 - \frac{2}{e} = 2\left( {1 - \frac{1}{e}} \right)$.

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