Answer
$$\frac{{{e^{3x}}}}{3} - 5{e^{ - x}} + C $$
Work Step by Step
$$\eqalign{
& \int {\left( {{e^{3x}} + 5{e^{ - x}}} \right)} dx \cr
& {\text{use sum rule for integration}} \cr
& = \int {{e^{3x}}} dx + \int {5{e^{ - x}}} dx \cr
& {\text{constant multiple rule}} \cr
& = \int {{e^{3x}}} dx + 5\int {{e^{ - x}}} dx \cr
& {\text{use the formula }}\int {{e^{ax}}} dx = \frac{{{e^{ax}}}}{a} + C \cr
& = \frac{{{e^{3x}}}}{3} + 5\left( {\frac{{{e^{ - x}}}}{{ - 1}}} \right) + C \cr
& {\text{simplifying}} \cr
& = \frac{{{e^{3x}}}}{3} - 5{e^{ - x}} + C \cr} $$
