Answer
$$2\ln \left| x \right| + 3{e^x} + C$$
Work Step by Step
$$\eqalign{
& \int {\left[ {\frac{2}{x} + 3{e^x}} \right]} dx \cr
& {\text{Split integrand}} \cr
& = \int {\frac{2}{x}} dx + \int {3{e^x}} dx \cr
& {\text{drop out constants}} \cr
& = 2\int {\frac{1}{x}} dx + 3\int {{e^x}} dx \cr
& {\text{Integrate using basic rules}} \cr
& = 2\ln \left| x \right| + 3{e^x} + C \cr
& \cr
& {\text{Checking by differentiation}} \cr
& \frac{d}{{dx}}\left[ {2\ln \left| x \right| + 3{e^x} + C} \right] \cr
& \frac{d}{{dx}}\left[ {2\ln \left| x \right|} \right] + \frac{d}{{dx}}\left[ {3{e^x}} \right] + \frac{d}{{dx}}\left[ C \right] \cr
& 2\frac{d}{{dx}}\left[ {\ln \left| x \right|} \right] + 3\frac{d}{{dx}}\left[ {{e^x}} \right] + \frac{d}{{dx}}\left[ C \right] \cr
& 2\left( {\frac{1}{x}} \right) + 3{e^x} + 0 \cr
& \frac{2}{x} + 3{e^x} \cr} $$
