Answer
$$2{e^{\sqrt x }} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{e^{\sqrt x }}}}{{\sqrt x }}dx} \cr
& = \int {{e^{\sqrt x }}\left( {\frac{1}{{\sqrt x }}} \right)dx} \cr
& {\text{substitute }}u = \sqrt x ,{\text{ }}du = \frac{1}{{2\sqrt x }}dx \cr
& \int {\frac{{{e^{\sqrt x }}}}{{\sqrt x }}dx} = 2\int {{e^u}du} \cr
& {\text{find the antiderivative}} \cr
& = 2{e^u} + C \cr
& {\text{replace back }}u = \sqrt x \cr
& = 2{e^{\sqrt x }} + C \cr} $$
