The value of _0^2 3^ x x d x is - Vidyadip

MCQ

The value of $\int_0^2 \frac{3^{\sqrt{x}}}{\sqrt{x}} d x$ is

✓
$\frac{2}{\log 3}\left(3^{\sqrt{2}}-1\right)$
B
$0$
C
$\frac{2 \sqrt{2}}{\log 3}$
D
$\frac{3^{\sqrt{2}}}{\sqrt{2}}$

✓

Answer

Correct option: A.

$\frac{2}{\log 3}\left(3^{\sqrt{2}}-1\right)$

(A)
Put $\sqrt{x}= t \Rightarrow \frac{1}{\sqrt{x}} d x=2 dt$
When $x=0, t =0$ and when $x=2, t =\sqrt{2}$
$\therefore \quad \int_0^2 \frac{3^{\sqrt{x}}}{\sqrt{x}} d x=2 \int_0^{\sqrt{2}} 3^{ t } dt =2\left[\frac{3^{ t }}{\log 3}\right]_0^{\sqrt{2}}=\frac{2}{\log 3}\left(3^{\sqrt{2}}-1\right)$

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