The value of _0^2 3^ x x ,dx is - Vidyadip

MCQ

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

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

✓

Answer

Correct option: A.

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

a
(a) Put $\sqrt x = t$ or $\frac{1}{{\sqrt x }}dx = 2dt$

Also, as $x = 0 $ to $2$ so, $t = 0$ to $\sqrt 2 $

Therefore, $\int_0^2 {\frac{{{3^{\sqrt x }}}}{{\sqrt x }}\,} dx $

$= 2\int_0^{\sqrt 2 } {{3^t}} dt $

$= 2\left[ {\frac{{{3^t}}}{{\log 3}}} \right]_0^{\sqrt 2 }$

$= \frac{2}{{\log 3}}({3^{\sqrt 2 }} - 1)$.

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