Answer
$$2\sqrt {{e^x}} + C$$
Work Step by Step
$$\eqalign{
& \int {\sqrt {{e^x}} } dx \cr
& {\text{Use }}\sqrt a = {a^{1/2}} \cr
& = \int {{{\left( {{e^x}} \right)}^{1/2}}} dx \cr
& {\text{Use the property }}{\left( {{a^m}} \right)^n} = {a^{mn}} \cr
& = \int {{e^{x/2}}} dx \cr
& {\text{Setting }}u = \frac{x}{2},\,\,\,\,\,\,\,\,du = \frac{1}{2}dx,\,\,\,\,\,\,2du = dx.{\text{ Thus}}{\text{,}} \cr
& \int {{e^{x/2}}} dx = \int {{e^u}} \left( 2 \right)du \cr
& = 2\int {{e^u}} du \cr
& {\text{Integrating }} \cr
& = 2{e^u} + C \cr
& {\text{substituting back }}u = \frac{x}{2} \cr
& = 2{e^{x/2}} + C \cr
& or \cr
& = 2\sqrt {{e^x}} + C \cr} $$
