TRANSFORM OF AN INTEGRAL
The transform of an integral can readily be found from that derived above for a derivative. Let the integral be
у — jx(t) dt
and let it be required to find y(s). By differentiating with respect to t, we get
EXTREME VALUE THEOREMS
Equation (2.3,14) may be rewritten as
—x(0) + sx(s) = I e~stx(t) dt rT
= lim e~stx(t) dt
T-* oo Jo
We now take the limit s —► 0 while T is held constant, i. e.
= lim x(t) dt = lim x(T) — a;(0)] T-* oo Jo T-* 00
This result, known as the final value theorem, provides a ready means for determining the asymptotic value of x(t) for large times from the value of its Laplace transform.
In a similar way, by taking the limit s —*■ oo at constant T, the integral vanishes for all finite x(t) and we get the initial value theorem.
lim sx(s) = x(0)