Table of Laplace Transforms

This is not a complete list of Laplace transforms, but it contains all common transforms, which can be used to quickly find solutions of differential equations and integrals:

$$${f{{\left({t}\right)}}}={{L}}^{{-{1}}}{\left({F}{\left({s}\right)}\right)}$$$	$$${F}{\left({s}\right)}={L}{\left({f{{\left({t}\right)}}}\right)}$$$
$$${1}$$$	$$$\frac{{1}}{{s}}$$$
$$${{t}}^{{n}}$$$, $$${n}={0},{1},{2},{3}\ldots$$$	$$$\frac{{{n}!}}{{{{s}}^{{{n}+{1}}}}}$$$
$$${{t}}^{{n}}$$$, $$${n}>-{1}$$$	$$$\frac{{\Gamma{\left({n}+{1}\right)}}}{{{s}}^{{{n}+{1}}}}$$$
$$${{e}}^{{{a}{t}}}$$$	$$$\frac{{1}}{{{s}-{a}}}$$$
$$${{t}}^{{{n}-\frac{{1}}{{2}}}}$$$, $$${n}={1},{2},{3}\ldots$$$	$$$\frac{{{1}\cdot{3}\cdot{5}\cdot\ldots\cdot{\left({2}{n}-{1}\right)}\cdot\sqrt{{\pi}}}}{{{{2}}^{{n}}{{s}}^{{{n}+\frac{{1}}{{2}}}}}}$$$
$$$\sqrt{{{t}}}$$$	$$$\frac{\sqrt{{\pi}}}{{{2}{{s}}^{{\frac{{3}}{{2}}}}}}$$$
$$${\sin{{\left({a}{t}\right)}}}$$$	$$$\frac{{a}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$$$
$$${\cos{{\left({a}{t}\right)}}}$$$	$$$\frac{{s}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$$$
$$${\sinh{{\left({a}{t}\right)}}}$$$	$$$\frac{{a}}{{{{s}}^{{2}}-{{a}}^{{2}}}}$$$
$$${\cosh{{\left({a}{t}\right)}}}$$$	$$$\frac{{s}}{{{{s}}^{{2}}-{{a}}^{{2}}}}$$$
$$${t}{\sin{{\left({a}{t}\right)}}}$$$	$$$\frac{{{2}{a}{s}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}}$$$
$$${t}{\cos{{\left({a}{t}\right)}}}$$$	$$$\frac{{{{s}}^{{2}}-{{a}}^{{2}}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}}$$$
$$${\sin{{\left({a}{t}+{b}\right)}}}$$$	$$$\frac{{{s}\cdot{\sin{{\left({b}\right)}}}+{a}\cdot{\cos{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$$$
$$${\cos{{\left({a}{t}+{b}\right)}}}$$$	$$$\frac{{{s}\cdot{\cos{{\left({b}\right)}}}-{a}\cdot{\sin{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$$$
$$${{e}}^{{{a}{t}}}{\sin{{\left({b}{t}\right)}}}$$$	$$$\frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}}$$$
$$${{e}}^{{{a}{t}}}{\cos{{\left({b}{t}\right)}}}$$$	$$$\frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}}$$$
$$${{e}}^{{{a}{t}}}{\sinh{{\left({b}{t}\right)}}}$$$	$$$\frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}}$$$
$$${{e}}^{{{a}{t}}}{\cosh{{\left({b}{t}\right)}}}$$$	$$$\frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}}$$$
$$${{t}}^{{n}}{{e}}^{{{a}{t}}}$$$, $$${n}={1},{2},{3}\ldots$$$	$$$\frac{{{n}!}}{{{\left({s}-{a}\right)}}^{{{n}+{1}}}}$$$
$$${f{{\left({c}{t}\right)}}}$$$	$$$\frac{{1}}{{c}}{F}{\left(\frac{{s}}{{c}}\right)}$$$
$$${u}_{{c}}{\left({t}\right)}={u}{\left({t}-{c}\right)}$$$	$$$\frac{{{e}}^{{-{c}{s}}}}{{s}}$$$
$$${u}_{{c}}{\left({t}\right)}{f{{\left({t}-{c}\right)}}}$$$	$$${{e}}^{{-{c}{s}}}{F}{\left({s}\right)}$$$
$$$\delta{\left({t}-{c}\right)}$$$	$$${{e}}^{{-{c}{s}}}$$$
$$${{e}}^{{{c}{t}}}{f{{\left({t}\right)}}}$$$	$$${F}{\left({s}-{c}\right)}$$$
$$${{t}}^{{n}}{f{{\left({t}\right)}}}$$$, $$${n}={1},{2},{3}\ldots$$$	$$${{\left(-{1}\right)}}^{{n}}{{F}}^{{{\left({n}\right)}}}{\left({s}\right)}$$$
$$${\int_{{0}}^{{t}}}{f{{\left(\tau\right)}}}{d}\tau$$$	$$$\frac{{{F}{\left({s}\right)}}}{{s}}$$$
$$${\int_{{0}}^{{t}}}{f{{\left({t}-\tau\right)}}}{g{{\left(\tau\right)}}}{d}\tau$$$	$$${F}{\left({s}\right)}{G}{\left({s}\right)}$$$
$$${f{'}}{\left({t}\right)}$$$	$$${s}{F}{\left({s}\right)}-{f{{\left({0}\right)}}}$$$
$$${f{''}}{\left({t}\right)}$$$	$$${{s}}^{{2}}{F}{\left({s}\right)}-{s}{f{{\left({0}\right)}}}-{f{'}}{\left({0}\right)}$$$
$$${{f}}^{{{\left({n}\right)}}}{\left({t}\right)}$$$	$$${{s}}^{{n}}{F}{\left({s}\right)}-{\sum_{{{k}={0}}}^{{{n}-{1}}}}{\left({{s}}^{{{n}-{1}-{k}}}{{f}}^{{{\left({k}\right)}}}{\left({0}\right)}\right)}$$$