# Engineering Tables/Laplace Transform Table

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