Laplace Transform Catalog

$x(t)$	$X(s)$	Region of Convergence
$\delta (t)$	$1$	All $s\in \mathbb{C}$
$u(t)$	$\frac{1}{s}$	$\text{Re}(s)>0$
$e^{-at}u(t)$	$\frac{1}{s+a}$	$\text{Re}(s)>-a$
$e^{at}u(t)$	$\frac{1}{s-a}$	$\text{Re}(s)>a$
$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$	$\text{Re}(s)>0$
$\cos ({\omega }_{0}t)u(t)$	$\frac{s}{s^2+{\omega }_0^2}$	$\text{Re}(s)>0$
$\sin ({\omega }_{0}t)u(t)$	$\frac{{\omega }_0}{s^2+{\omega }_0^2}$	$\text{Re}(s)>0$
$e^{-at}\cos (\omega t)u(t)$	$\frac{s+a}{(s+a{)}^2+{\omega }^2}$	$\text{Re}(s)>-a$
$e^{-at}\sin (\omega t)u(t)$	$\frac{\omega }{(s+a{)}^2+{\omega }^2}$	$\text{Re}(s)>-a$
$\delta (t-T)$	$e^{-sT}$	All $s\in \mathbb{C}$
$u(-t)$	$-\frac{1}{s}$	$\text{Re}(s)<0$

All of these have a $u(t)$ factor because we assume casual signals