Engineering Tables/Laplace Transform Table

	Time Domain	Laplace Domain
	$x(t) = \mathcal{L}^{-1}\left\{ X(s) \right\}$	$X(s) = \mathcal{L} \left\{ x(t) \right\}$
1	$\frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s)e^{st}ds$	$\int_{-\infty}^\infty x(t)e^{-st}dt$
2	$\delta (t) \,$	$1 \,$
3	$\delta (t-a)\,$	$e^{-as}\,$
4	$u(t) \,$	$\frac{1}{s}$
5	$u(t-a)\,$	$\frac{e^{-as}}{s}$
6	$t u(t) \,$	$\frac{1}{s^2}$
7	$t^nu(t) \,$	$\frac{n!}{s^{n+1}}$
8	$\frac{1}{\sqrt{\pi t}}u(t)$	$\frac{1}{\sqrt{s}}$
9	$e^{at}u(t) \,$	$\frac{1}{s-a}$
10	$t^n e^{at}u(t) \,$	$\frac{n!}{(s-a)^{n+1}}$
11	$\cos (\omega t) u(t) \,$	$\frac{s}{s^2+\omega^2}$
12	$\sin (\omega t) u(t) \,$	$\frac{\omega}{s^2+\omega^2}$
13	$\cosh (\omega t) u(t) \,$	$\frac{s}{s^2-\omega^2}$
14	$\sinh (\omega t) u(t) \,$	$\frac{\omega}{s^2-\omega^2}$
15	$e^{at} \cos (\omega t) u(t) \,$	$\frac{s-a}{(s-a)^2+\omega^2}$
16	$e^{at} \sin (\omega t) u(t) \,$	$\frac{\omega}{(s-a)^2+\omega^2}$
17	$\frac{1}{2\omega^3}(\sin \omega t-\omega t \cos \omega t)$	$\frac{1}{(s^2+\omega^2)^2}$
18	$\frac{t}{2\omega} \sin \omega t$	$\frac{s}{(s^2+\omega^2)^2}$
19	$\frac{1}{2\omega}(\sin \omega t+\omega t \cos \omega t)$	$\frac{s^2}{(s^2+\omega^2)^2}$