Question

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

Answer 1

$ℒ(f(t))=\int_0^{\infty}e^{-st}f(t) dt=$

$=\int_0^1e^{-st}t dt+\int_1^{\infty}e^{-st}dt$

$\int_0^1e^{-st}tdt=-\frac{1}{s}e^{-st}t|_0^1+\int_0^1\frac{e^{-st}}{s}dt=$

$=-\frac{1}{s}e^{-s}-\frac{1}{s^2}e^{-st}|_0^1=$

$=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)$

$\int_1^{\infty}e^{-st}dt=-\frac{1}{s}e^{-st}|_1^{\infty}=\frac{1}{s}e^{-s}$

$ℒ(f(t))=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)+\frac{1}{s}e^{-s}=$

$=\frac{1}{s^2}(1-e^{-s})$