Engineering Tables/Laplace Transform Table

No.	Time Domain $x(t)={ℒ}^{-1}\{X(s)\}$	Laplace Domain $X(s)=ℒ\{x(t)\}$
1	$\frac{1}{2\pi j}\underset{\sigma -j\mathrm{\infty }}{\overset{\sigma +j\mathrm{\infty }}{\int }}X(s)e^{st}ds<math>|<math>\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-st}dt$
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	$tu(t)$	$\frac{1}{s^2}$
7	$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$
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	$\frac{1}{2\omega }(\sin \omega t+\omega t\cos \omega t)$	$\frac{s^2}{(s^2+{\omega }^2{)}^2}$