Answer
$$2{e^x} + \frac{3}{2}{e^{ - 2x}} + C $$
Work Step by Step
$$\eqalign{
& \int {\left( {2{e^x} - 3{e^{ - 2x}}} \right)} dx \cr
& {\text{use sum rule for integration}} \cr
& = \int {2{e^x}} dx - \int {3{e^{ - 2x}}} dx \cr
& {\text{constant multiple rule}} \cr
& = 2\int {{e^x}} dx - 3\int {{e^{ - 2x}}} dx \cr
& {\text{use the formula }}\int {{e^{ax}}} dx = \frac{{{e^{ax}}}}{a} + C \cr
& = 2{e^x} - 3\left( {\frac{{{e^{ - 2x}}}}{{ - 2}}} \right) + C \cr
& {\text{simplifying}} \cr
& = 2{e^x} + \frac{3}{2}{e^{ - 2x}} + C \cr} $$
