Engineering Tables/Laplace Transform Table

No.	Time Domain $x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$	Laplace Domain $X(s)={\mathcal {L}}\left\{x(t)\right\}$
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	$tu(t)\,$	${\frac {1}{s^{2}}}$
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	${\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)$	${\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}$