Answer
\[ = \frac{1}{{s - a}}\]
Work Step by Step
\[\begin{gathered}
f\,\left( t \right) = {e^{et}}\,\,\,\,\,\, \to \,\,\,\,\,f\,\left( s \right) = \frac{1}{{s - a}} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\,\mathcal{L}\,\left[ {{e^{at}}u\,\left( t \right)} \right] = \int_0^\infty {{e^{at}}{e^{st}}\,dt} \hfill \\
\hfill \\
multiply\,\,and\,\,integrate \hfill \\
\hfill \\
= \left. {\frac{1}{{s - a}}\,\left( {s - a} \right)t} \right|_0^\infty \hfill \\
\hfill \\
use\,\,\,the\,\,ftc \hfill \\
\hfill \\
= \frac{1}{{s - a}} \hfill \\
\end{gathered} \]