The correct option is B 2 (1 1e) We know, #160;By parts integrals ∫ u.v dx=u∫ v dx ∫ ( dudx∫ v dx )dx ∫ lnx · 1dx = lnx∫ dx ∫1/x∫ dx dx ...

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Property 2

∫1/ee|ln x|dx...

Question

$\int_{1 / e}^{e} | ln x | d x$ equals

e^{- 1} - 1

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2 (1 - \frac{1}{e})

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1 - \frac{1}{e}

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e - 1

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Solution

The correct option is B $2 (1 - \frac{1}{e})$
We know,
By parts integrals
$\int u . v d x = u \int v d x$ $- \int (\frac{d u}{d x} \int v d x) d x$

$\int l n x \cdot 1 d x = l n x \int d x - \int \frac{1}{x} \int d x d x$

$= x l n x - x + c$

$\int_{1 / e}^{e} | l n x | d x = \int_{1 / e}^{1} | l n x | d x + \int_{1}^{e} | l n x | d x$

$= \int_{1 / e}^{- 1} l n x d x + \int_{1}^{e} l n x d x$

$= - [x (l n x - 1)]_{1 / e}^{1} + [x | l n x - 1 |]_{1}^{e}$

$= - [- 1 - \frac{1}{e} (- 2)] + [0 - 1 (- 1)]$

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

$∴$ Option B is correct

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