Answer
$$2[e^2-e]$$
Work Step by Step
Given $$ \int_{1}^{4}\frac{e^{\sqrt{t}}}{\sqrt{t}} dt$$
Let $u=\sqrt{t}\ \ \ \Rightarrow \ \ \ \dfrac{1}{2\sqrt{t}}dt$, at $x=1\to u=1$ at $x=4\to u=2$then
\begin{align*}
\int_{1}^{4}\frac{e^{\sqrt{t}}}{\sqrt{t}} dt&=2 \int_{1}^{2} e^{u} du\\
&=2e^u\bigg|_{1}^{2}\\
&=2[e^2-e]
\end{align*}