Laplace transform of integral
On can show that if a real function isLaplace-transformable (http://planetmath.org/LaplaceTransform), as wellis . The latter is alsocontinuous for and by theNewton–Leibniz formula (http://planetmath.org/FundamentalTheoremOfCalculus),has the derivative equal . Hence we may apply theformula for Laplace transform of derivative, obtaining
i.e.
(1) |
Application. We start from the easily derivable rule
where the curved from the Laplace-transformed function to the original function. The formula (1) thus yields successively
etc. Generally, one has
(2) |