Use Definition 7.1.1.
DEFINITION 7.1.1 Laplace Transform
Let f be a function defined for t ≥ 0.
Then the integral
ℒ{f(t)} = ∞
e−stf(t) dt0
is said to be the Laplace transform of f, provided that the integral converges.
Find ℒ{f(t)}.
(Write your answer as a function of s.)
f(t) = {t, 0 ≤ t < 1
{1, t ≥ 1
L(f(t))=∫0∞e−stf(t)dt=ℒ(f(t))=\int_0^{\infty}e^{-st}f(t) dt=L(f(t))=∫0∞e−stf(t)dt=
=∫01e−sttdt+∫1∞e−stdt=\int_0^1e^{-st}t dt+\int_1^{\infty}e^{-st}dt=∫01e−sttdt+∫1∞e−stdt
∫01e−sttdt=−1se−stt∣01+∫01e−stsdt=\int_0^1e^{-st}tdt=-\frac{1}{s}e^{-st}t|_0^1+\int_0^1\frac{e^{-st}}{s}dt=∫01e−sttdt=−s1e−stt∣01+∫01se−stdt=
=−1se−s−1s2e−st∣01==-\frac{1}{s}e^{-s}-\frac{1}{s^2}e^{-st}|_0^1==−s1e−s−s21e−st∣01=
=−1se−s−1s2(e−s−1)=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)=−s1e−s−s21(e−s−1)
∫1∞e−stdt=−1se−st∣1∞=1se−s\int_1^{\infty}e^{-st}dt=-\frac{1}{s}e^{-st}|_1^{\infty}=\frac{1}{s}e^{-s}∫1∞e−stdt=−s1e−st∣1∞=s1e−s
L(f(t))=−1se−s−1s2(e−s−1)+1se−s=ℒ(f(t))=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)+\frac{1}{s}e^{-s}=L(f(t))=−s1e−s−s21(e−s−1)+s1e−s=
=1s2(1−e−s)=\frac{1}{s^2}(1-e^{-s})=s21(1−e−s)
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