Evaluate: _0^1 e ^x e ^x-1 d x - Vidyadip

Question

Evaluate: $\int_0^1 \frac{ e ^x}{\sqrt{ e ^x-1}} d x$

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Answer

$\int_0^1 \frac{ e ^x}{\sqrt{ e ^x-1}} d x=\left[2 \sqrt{ e ^x-1}\right]_0^1 \quad \ldots \ldots . .\left[\because \frac{ f ^{\prime}(x)}{\sqrt{ f (x)}} d x=2 \sqrt{ f (x)}+ c \right]$
$=2\left(\sqrt{ e ^1 1}-\sqrt{ e ^0-1}\right)$
$=2(\sqrt{ e -1}-\sqrt{1-1})$
$=2 \sqrt{ e -1}$

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