Question

Solve

${\int }_0^2\frac{3^{\sqrt{x}}}{\sqrt{x}}dx$, equals-

• A. $\frac{2}{ln3}(3^{\sqrt{2}}-1)$
• B. $0$
• C. $\frac{2\sqrt{2}}{ln3}$
• D. $\frac{3^{\sqrt{2}}}{\sqrt{2}}$

Answer 1

The correct option is A $\frac{2}{ln3}(3^{\sqrt{2}}-1)$

${\int }_0^2\frac{3^{\sqrt{x}}}{\sqrt{x}}dx$

Let $t=\sqrt{x}\Rightarrow dt=\frac{1}{2\sqrt{x}}dx$

When $x=0\Rightarrow t=0$

When $x=2\Rightarrow t=\sqrt{2}$

${\int }_0^2\frac{3^{\sqrt{x}}}{\sqrt{x}}dx$

$=2{\int }_0^{\sqrt{2}}{3}^{t}dt$

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

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

$=\frac{2}{\ln 3}[3^{\sqrt{2}}-3^0]$

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