Laplace transform of f(t)t

Suppose that the quotient


is Laplace-transformable (  It follows easily that also f(t) is such.  According to the parent entry (, we may write





G(s)=-F(-1)(s)+C (1)

where F(-1)(s) means any antiderivative of F(s).  Since each Laplace transformed functionMathworldPlanetmath vanishes in the infinity  s=  and thus  G()=0,  the equation (1) implies


and therefore


We have obtained the result

{f(t)t}=sF(u)𝑑u. (2)

Application.  By the table of Laplace transformsDlmfMathworldPlanetmath,  {sint}=1s2+1.  Accordingly the formula (2) yields


Thus we have

{sintt}=arccots=arctan1s. (3)

This result is derived in the entry Laplace transform of sine integral in two other ways.

Title Laplace transform of f(t)t
Canonical name LaplaceTransformOffracftt
Date of creation 2014-03-08 15:45:15
Last modified on 2014-03-08 15:45:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 44A10
Related topic FundamentalTheoremOfCalculusClassicalVersion
Related topic SubstitutionNotation
Related topic CyclometricFunctions