If _ 2 ^a e ^x e ^x-1 d x=2, then a = - Vidyadip

MCQ

If $\int_{\log 2}^a \frac{ e ^x}{\sqrt{ e ^x-1}} d x=2$, then $a =$

A
$0$
B
2 log 2
✓
log 5
D
log 2

✓

Answer

Correct option: C.

log 5

(C)
$\int_{\log 2}^{ a } \frac{ e ^x}{\sqrt{ e ^x-1}} d x=2$
Put $e ^x-1= t \Rightarrow e ^x d x= dt$
$\therefore \quad \int_1^{ e ^{ a }-1} \frac{ dt }{\sqrt{ t }}=2$
$\Rightarrow[2 \sqrt{ t }]_1^{ e ^{ a }-1}=2$
$\Rightarrow \sqrt{ e ^{ a }-1}-1=1$
$\Rightarrow \sqrt{ e ^{ a }-1}=2$
$\Rightarrow e ^{ a }-1=4$
$\Rightarrow e ^{ a }=5$
$\Rightarrow a =\log 5$

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