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The Laplace transform of a continuous function over the interval $[0, \infty)$ is defined by $F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$ (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of $F$ is the set of all real numbers $s$ such that the improper integral converges. Find the Laplace transform $F$ of each of the following functions and give the domain of $F .$$$f(x)=x$$

$F(s)=\frac{1}{s^{2}}$Domain of $F$ is $(0, \infty)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 3

Techniques of Integration

Section 7

Improper Integrals

Integration

Integration Techniques

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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So when this question we know that we cannot we write our function half of us for this boss transformation with a continuous function over the intervals your infinity as this integral infinity zero. And then here we have either the power of negative at Saks. This is from the formula. And then here we put effort backs, which in our case, is just gonna be acts. And then dx the box is given. And the question is, now we can rewrite. This is the limit. The limit is two goes to infinity of our interval t zero, actually the power of negative at Saks Steeples. Over. Wish we cannot write. Rewrite. This portion is negative. One over a squared, even power of negative ass axe one boss ass Fax this entire thing. Use your So now this portion again, we're keeping this limited t goes to infinity. We can make this into one minus either the power of negative Esty Times one plus Ask Steve And then we find now that F Vass is just gonna be equal and for this portion right here. But actually, we're gonna want to put the limited's t goes to infinity. So it's just this limit. And then Juan over Escort is on the outside in this case. So when we evaluate this limit, you realize that after bus, and, um, in the end, it's just gonna be one of grass squared. And the domain about the best is gonna be zero to infinity, not including zero. So these are our final answers to this question.

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