Answer
\[ = \frac{1}{s}\]
Work Step by Step
\[\begin{gathered}
f\,\left( t \right) = 1\,\,\,\, \to \,\,f\,\left( s \right) = \frac{1}{s} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\mathcal{L} = \int_0^\infty {{e^{ - st}}\,\left( 1 \right)dt} \hfill \\
\hfill \\
\mathcal{L} = \mathop {\lim }\limits_{b \to \infty } \int_0^b {{e^{ - st}}\,dt} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
\mathcal{L} = \left. {\mathop {\lim }\limits_{b \to \infty } \frac{{ - {e^{ - st}}}}{s}} \right|_0^b \hfill \\
\hfill \\
use\,\,the\,\,ftc \hfill \\
\hfill \\
= \mathop {\lim }\limits_{b \to \infty } \frac{{ - {e^{sb}} + 1}}{s} \hfill \\
\hfill \\
evaluate\,\,the\,\,limit \hfill \\
\hfill \\
= \frac{1}{s} \hfill \\
\end{gathered} \]
