Laplace transform

Basic pairs

$ f(t) $	$ F(s) = \mathcal{L}\{f(t)\} $	Region of application
$ a $ (constant)	$ \dfrac{a}{s} $	$ s>0 $
$ t $	$ \dfrac{1}{s^{2}} $	$ s>0 $

$$ \mathcal{L}\{2 + 5t\} = 2\,\mathcal{L}\{1\} + 5\,\mathcal{L}\{t\} $$

$$ = 2\,\frac{1}{s} \; + \; 5\,\frac{1}{s^{2}} = \frac{2}{s} + \frac{5}{s^{2}} $$

Note. The stated regions $s>0$ correspond to the conditions under which the defining integrals converge, i.e., $\operatorname{Re}(s)>0$.

\( f(t) \)	\( F(s) = \mathcal{L}\{f(t)\} \)	Region of application
\( a \) (constant)	\( \dfrac{a}{s} \)	\( s>0 \)
\( t \)	\( \dfrac{1}{s^{2}} \)	\( s>0 \)