The Laplace transform of an integral involving a function
(
)
f(t) is given by a property known as the "Differentiation Theorem" or "Shift Theorem" of Laplace transforms. The theorem states that if
(
)
f(t) is a piecewise continuous function on the interval
[
0
,
â
)
[0,â) and
(
)
F(s) is its Laplace transform, then the Laplace transform of the integral of
(
)
f(t) is related to
(
)
F(s) as follows:
{
âĢ
0
(
)
â
}
=
(
)
L{âĢ
0
t
â
f(Ī)dĪ}=
s
F(s)
â
In mathematical notation:
{
âĢ
0
(
)
â
}
=
(
)
L{âĢ
0
t
â
f(Ī)dĪ}=
s
F(s)
â
Where:
L represents the Laplace transform operator.
(
)
f(t) is the function being integrated.
(
)
F(s) is the Laplace transform of
(
)
f(t).
s is the complex frequency parameter.
This theorem is very useful when dealing with Laplace transforms of integrals in circuit analysis and control systems engineering, where it helps simplify the calculation of Laplace transforms of time-domain functions involving integrals.