Answer
\[ = - \frac{4}{3}{e^{ - 3x}} - \frac{1}{5}{e^{ - 5x}} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{4 + {e^{ - 2x}}}}{{{e^{3x}}}}dx} \hfill \\
\hfill \\
{\text{split}}\,\,{\text{the}}\,\,{\text{integrand}} \hfill \\
\hfill \\
= \int_{}^{} {\,\left( {\frac{4}{{{e^{3x}}}} + \frac{{{e^{ - 2x}}}}{{{e^{3x}}}}} \right)dx} \hfill \\
\hfill \\
{\text{Simplify}} \hfill \\
\hfill \\
= \int_{}^{} {\,\left( {4{e^{ - 3x}} + {e^{ - 5x}}} \right)} dx \hfill \\
\hfill \\
\operatorname{int} egrating \hfill \\
\hfill \\
= - \frac{4}{3}{e^{ - 3x}} - \frac{1}{5}{e^{ - 5x}} + C \hfill \\
\end{gathered} \]