Answer
$$\frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {{e^{2t}} + 2\sqrt t } \right)} dt \cr
& {\text{use radical property}} \cr
& = \int {\left( {{e^{2t}} + 2{t^{1/2}}} \right)} dt \cr
& {\text{split the integrand}} \cr
& = \int {{e^{2t}}} dt + \int {2{t^{1/2}}} dt \cr
& {\text{integrate}} \cr
& = \frac{1}{2}\left( {{e^{2t}}} \right) + 2\left( {\frac{{{t^{3/2}}}}{{3/2}}} \right) + C \cr
& = \frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dt}}\left( {\frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C} \right) \cr
& {\text{ = }}\frac{d}{{dt}}\left( {\frac{1}{2}{e^{2t}}} \right) + \frac{d}{{dt}}\left( {\frac{4}{3}{t^{3/2}}} \right) + \frac{d}{{dt}}\left( C \right) \cr
& {\text{ = }}\frac{1}{2}\left( {{e^{2t}}} \right)\left( 2 \right) + \frac{4}{3}\left( {\frac{3}{2}} \right){t^{1/2}} + 0 \cr
& = {e^{2t}} + 2\sqrt t \cr} $$
