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# If $f(t)$ is continuous for $t \ge 0$, the Laplace transform of $f$ is the function $F$ defined by$$F(s) = \int_0^\infty f(t) e^{-st}\ dt$$and the domain of $F$ is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions.(a) $f(t) = 1$ (b) $f(t) = e^t$ (c) $f(t) = t$

## a. $F(s)=\frac{1}{s}$ with domain $\{s | s>0\}$b. $F(s)=\frac{1}{s-1}$ with domain $\{s | s>1\}$c. $F(s)=\frac{1}{s^{2}}$ and the domain of $F$ is $\{s | s>0\}$

#### Topics

Integration Techniques

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

if FT is continuous for tea is greater than zero. The lap last transform off. Off. It's a function, captain. Half defined by bucks. Yes, he got to interrupt from zero to twenty after house later. Negative as stated. And is that the way off? Half is a such considering off our numbers for which the integral converters I find that the low class transforms the foreign functions A after Izzy or wine Hey of us. Yes, we caught you. Name it. A cost to infinity. Into girl from zero to I in connective Castillo. Thank you. This it is. He caught you, Lim. A costume affinity. Negative one over us. You too. Negative asked house, eh? Minus one. This's the culture. One over us. Years at the Wayne is US authorities are zero. How to be lefty is you got to eat the power off. Yes, It's the conscious that limit a cost to your vanity. Integral from zero to a You two one one of us. Yeah. This is about you claim it. A cost to vanity won over one of us terms. I need to make you one. It's you one minus asked. House, eh? Minus one This is equal to one over one negative one over one. Dewayne is as great is on one part of C. After you see on TV, then we have a CE That's the limit. A cost three affinity Integral Found zero to a he has connective. As to you Here we use integration. My parts you hear the result is lim a cost to infinity. A hams, Negative one over us into a negative times. A minus one over us square halves. Each connective asked house, eh? Minus one If it is a good time. Zero we have limit of this part is equal to zero. And this this part is you got zero. So the answer is one over us square. Here's what the mind is as the secret is on zero.

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#### Topics

Integration Techniques

##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

Lectures

Join Bootcamp