Laplace transform of integral

On can show that if a real functiontf(t)  is Laplace-transformable (, as well is 0tf(τ)𝑑τ.  The latter is also continuousMathworldPlanetmath for  t>0  and by the Newton–Leibniz formula (, has the derivative equal f(t).  Hence we may apply the formula for Laplace transform of derivative, obtaining



{0tf(τ)𝑑τ}=F(s)s. (1)

Application.  We start from the easily derivable rule

1s 1,

where the curved from the Laplace-transformed functionMathworldPlanetmath to the original function.  The formula (1) thus yields successively


etc.  Generally, one has

1sntn-1(n-1)!n+. (2)
Title Laplace transform of integral
Canonical name LaplaceTransformOfIntegral
Date of creation 2014-03-17 10:43:31
Last modified on 2014-03-17 10:43:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 44A10