 # Table of Laplace Transforms

## Definition of Laplace Transforms

Let f(t) be a function of the real variable t, such that t ≥ 0. The
Laplace transform F(s) of f is given by the integral
F(s) = L(f(t) = 0 e -st f(t) dt

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 1 / s t 1 / s 2 t n n! / s n+1 , (n = 1,2,3...) t 1/2 Pi 1/2 / 2s 3/2 t -1/2 (Pi / s) 1/2 e -a t 1 / (s + a) t e -a t 1 / (s + a) 2 sin a t a / (s 2 + a 2) t sin a t 2 a s / (s 2 + a 2) 2 e -at sin b t b / ( (s + a) 2 + b 2 ) cos a t s / (s 2 + a 2) t cos a t (s 2 - a 2) / (s 2 + a 2) 2 e -at cos b t (s + a) / ( (s + a) 2 + b 2 ) sinh a t a / (s 2 - a 2) cosh a t s / (s 2 - a 2) 1 - cos a t a 2 / ( s (s 2 + a 2) ) (2 / t)( t - cos a t) ln [ (s 2 + a 2) / s 2 ] (2 / t)( t - cosh a t) ln [ (s 2 - a 2) / s 2 ] (1 / t)( sin a t) arctan(a / s)

## More references on integrals in calculus

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