Question

The value of $\underset{0}{\overset{2}{\int }}\frac{3^{\sqrt{x}}}{\sqrt{x}}dx$ is ________________.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\underset{0}{\overset{2}{\int }}\frac{3^{\sqrt{x}}}{\sqrt{x}}dx \\ \mathrm{Let}\sqrt{x}=t \\ \Rightarrow d\left(\sqrt{x}\right)=dt \\ \Rightarrow \frac{1}{2\sqrt{x}}dx=dt \\ \mathrm{Also},\mathrm{if}x=0,t=0 \\ \mathrm{if}x=2,t=\sqrt{2} \\ \mathrm{Thus}, \\ I=\underset{0}{\overset{\sqrt{2}}{\int }}{3}^t\left(2dt\right) \\ =2\underset{0}{\overset{\sqrt{2}}{\int }}{3}^{t}dt \\ =2{\left[\frac{3^t}{\log 3}\right]}_0^{\sqrt{2}} \\ =2\left[\frac{3^{\sqrt{2}}}{\log 3}-\frac{3^0}{\log 3}\right] \\ =2\left[\frac{3^{\sqrt{2}}-1}{\log 3}\right] \end{gathered}$$

​Hence, the value of $\underset{0}{\overset{2}{\int }}\frac{3^{\sqrt{x}}}{\sqrt{x}}dx$ is $\underline{2\left[\frac{3^{\sqrt{2}}-1}{\log 3}\right]}$.