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Table of Laplace Transforms

Definition of Laplace Transforms

Let

$f(t)$ be a function of the real variable

$t$ , such that

$t \geq 0$ . The Laplace transform

$F(s)$ of

$f$ is given by the integral

$F(s) = L(f(t) = \int_{0}^{\infty} e^{-st} f(t) \, dt$ where

$s$ is a complex variable.

$f(t)$ is called the original and

$F(s)$ is called the image function.

Table of Laplace Transforms

$f(t)$	$F(s)$
1	$\dfrac{1}{s}$
$t$	$\dfrac{1}{s^2}$
$t^n$	$\dfrac{n!}{s^{n+1}}$ , $(n = 1,2,3...)$
$t^{1/2}$	$\dfrac{\sqrt{\pi}}{2s^{3/2}}$
$t^{-1/2}$	$\sqrt{\dfrac{\pi}{s}}$
$e^{-at}$	$\dfrac{1}{s + a}$
$t e^{-at}$	$\dfrac{1}{(s + a)^2}$
$\sin(at)$	$\dfrac{a}{s^2 + a^2}$
$t \sin(at)$	$\dfrac{2as}{(s^2 + a^2)^2}$
$e^{-at} \sin(bt)$	$\dfrac{b}{(s + a)^2 + b^2}$
$\cos(at)$	$\dfrac{s}{s^2 + a^2}$
$t \cos(at)$	$\dfrac{s^2 - a^2}{(s^2 + a^2)^2}$
$e^{-at} \cos(bt)$	$\dfrac{s + a}{(s + a)^2 + b^2}$
$\sinh(at)$	$\dfrac{a}{s^2 - a^2}$
$\cosh(at)$	$\dfrac{s}{s^2 - a^2}$
$1 - \cos(at)$	$\dfrac{a^2}{s(s^2 + a^2)}$
$\left(\dfrac{2}{t}\right)(t - \cos(at))$	$\ln \left( \dfrac{s^2 + a^2}{s^2} \right)$
$\left(\dfrac{2}{t}\right)(t - \cosh(at))$	$\ln \left( \dfrac{s^2 - a^2}{s^2} \right)$
$\left(\dfrac{1}{t}\right)(\sin(at))$	$\arctan \left( \dfrac{a}{s} \right)$

More references on integrals in calculus

Laplace Transform with Examples and Solutions
Find Area Under Curve
table of integrals
properties of integrals