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) dt 0 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.)

Answer 1

Answer:

$F(s)=\frac{1}{s}$

Step-by-step explanation:

Let $f(t)$ be defined for $t\ge0$.

The Laplace transform of $f(t)$ is an integral transform given by the Laplace integral:

$\ell(f(t))=F(s)=\int\limits^{\infty}_0 {e^{-st}}\cdot f(t) \, dt$, provided that this improper integral exists.

If we let $f(t)=1$, then

$\ell(f(t))=F(s)=\int\limits^{\infty}_0 {e^{-st}}\cdot1 \, dt$

$\ell(f(t))=F(s)=\lim_{a\to \infty}\int\limits^{a}_0 {e^{-st}} dt$

$F(s)=\lim_{a\to \infty}-\frac{1}{s} {e^{-st}}|_{0}^{a}$

$F(s)=-\frac{1}{s} {\lim_{a\to \infty}e^{-as}}--\frac{1}{s} {\lim_{a\to \infty}e^{-0\times s}}$

Recall that: $\lim_{x\to \infty}e^{-x}=0$

$\implies F(s)=0+\frac{1}{s}$

$\implies F(s)=\frac{1}{s}$