If - 2 x e x -1 -1 dx- 3 2 then x-

The correct option is D e^2 According to, [ ∫ #160; _log #160; 2 ^ x ( e^ x 1) ^ 1 dx= log #160; ...

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Standard XII

Mathematics

Integration Using Substitution

If ∫ 2 x e...

Question

If $\int_{2}^{x} {(e^{x} - 1)}^{- 1} d x = (\frac{3}{2})$ then $x =$

ln 4

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1

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e^{2}

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\frac{1}{e}

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Solution

The correct option is D $e^{2}$
According to,
$\begin{matrix} \int_{log 2}^{x} {(e^{x} - 1)}^{- 1} d x = log (\frac{3}{2}) \int_{log 2}^{x} \frac{1}{(e^{x} - 1)} d x = log (\frac{3}{2}) \int_{log 2}^{x} \frac{e^{- x}}{(1 - e^{- x})} d x = log (\frac{3}{2}) {[log (1 - e^{- x})]}_{log 2}^{x} = log (\frac{3}{2}) log (1 - e^{- x}) - l o g \frac{1}{2} = log (\frac{3}{2}) log (1 - e^{- x}) = log (\frac{3}{4}) (1 - e^{- x}) = (\frac{3}{4}) e^{- x} = \frac{1}{4} e^{x} = 4 = 2^{2} ∴ x = 2 t h e n, e^{2} \end{matrix}$
Sothe correct option is C.

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If ∫x2(ex−1)−1dx=(32) then x=

The correct option is D e2According to,∫xlog2(ex−1)−1dx=log(32)∫xlog21(ex−1)dx=log(32)∫xlog2e−x(1−e−x)dx=log(32)[log(1−e−x)]xlog2=log(32)log(1−e−x)−log12=log(32)log(1−e−x)=log(34)(1−e−x)=(34)e−x=14ex=4=22∴x=2then,e2Sothe correct option is C.

If $\int_{2}^{x} {(e^{x} - 1)}^{- 1} d x = (\frac{3}{2})$ then $x =$