Engineering Tables/Laplace Transform Table

Time Domain	Laplace Domain
$x(t)=\frac{1}{2\pi j}\underset{\sigma -j\mathrm{\infty }}{\overset{\sigma +j\mathrm{\infty }}{\int }}X(s)e^{st}ds$	$X(s)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-st}dt$
$\delta (t)$	$1$
$\delta (t-a)$	$e^{-as}$
$u(t)$	$\frac{1}{s}$
$u(t-a)$	$\frac{e^{-as}}{s}$
$tu(t)$	$\frac{1}{s^2}$
$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$
$\frac{1}{\sqrt{\pi t}}u(t)$	$\frac{1}{\sqrt{s}}$
$e^{at}u(t)$	$\frac{1}{s-a}$
$t^n{e}^{at}u(t)$	$\frac{n!}{(s-a{)}^{n+1}}$
$\cos (\omega t)u(t)$	$\frac{s}{s^2+{\omega }^2}$
$\sin (\omega t)u(t)$	$\frac{\omega }{s^2+{\omega }^2}$
$\cosh (\omega t)u(t)$	$\frac{s}{s^2-{\omega }^2}$
$\sinh (\omega t)u(t)$	$\frac{\omega }{s^2-{\omega }^2}$
$e^{at}\cos (\omega t)u(t)$	$\frac{s-a}{(s-a{)}^2+{\omega }^2}$
$e^{at}\sin (\omega t)u(t)$	$\frac{\omega }{(s-a{)}^2+{\omega }^2}$
$\frac{1}{2{\omega }^3}(\sin \omega t-\omega t\cos \omega t)$	$\frac{1}{(s^2+{\omega }^2{)}^2}$
$\frac{t}{2\omega }\sin \omega t$	$\frac{s}{(s^2+{\omega }^2{)}^2}$
$\frac{1}{2\omega }(\sin \omega t+\omega t\cos \omega t)$	$\frac{s^2}{(s^2+{\omega }^2{)}^2}$