Table of Laplace Transforms of Elementary Functions

Below are some functions   $f(t)$   and their Laplace transforms   $F(s)$.
 

$f(t)$ $F(s) = \mathcal{L} \left\{f(t)\right\}$
$1$ $\dfrac{1}{s}$
$t$ $\dfrac{1}{s^2}$
$t^2$ $\dfrac{2}{s^3}$
$t^n, \, (n = 1, \, 2, \, 3\, ...)$ $\dfrac{n!}{s^{n + 1}}$
$t^a, \, (a \, \text{ is positive})$ $\dfrac{\Gamma(a + 1)}{s^{a + 1}}$
$e^{at}$ $\dfrac{1}{s - a}$
$\sin bt$ $\dfrac{b}{s^2 + b^2}$
$\cos bt$ $\dfrac{s}{s^2 + b^2}$
$\sinh at$ $\dfrac{a}{s^2 - a^2}$
$\cosh at$ $\dfrac{s}{s^2 - a^2}$

 

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