Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

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 .$$$f(x)=e^{a x}$$

$F(s)=\frac{1}{s-a}$Domain of $F$ is $(a, \infty)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 3

Techniques of Integration

Section 7

Improper Integrals

Integration

Integration Techniques

Missouri State University

Campbell University

University of Nottingham

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:40

The Laplace transform of a…

03:19

08:52

03:40

If $ f(t) $ is continuous …

01:24

In Exercises $86-89$ , the…

02:04

Use properties of the Lapl…

02:46

A powerful tool in solving…

06:26

04:05

Suppose that $ 0 \le f(t) …

05:15

So in this question, you know that the loss transformer continuous function can be rewritten. As this is the interval from zero to infinity, including syrup is gonna be eating the power of negative at Saks. And then here we put our function, which, in our case, is eating the power of a axe. Not yet. So now we're gonna be right. This is the limit, which is? The limit is t goes to infinity of and then this is gonna be t and your infinity hair T is going to infinity Sweep a t here instead now And you The power, eh? Minus ass times Acts and and d s, which can now be recognized one over. We're reading this worse. You keeping the limit? One over a minus. Fast times either power. But they might just ask. That's you're okay. So now we can rewrite again, keeping this limited. Steve goes to infinity, will rewrite this portion. It's one over. A minus ass tries. You do the power being minus past acts. You're a minus s t now minus one over a minus s, which evaluate. Now we're violating the limit. It's well, it's just gonna be won over asked me to say so. This is actually our function after us. It's gonna be one of her ass minus a. And our domain is gonna be a to infinity. So these right here are finally answers to this question.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

The Laplace transform of a continuous function over the interval $[0, \infty…

If $ f(t) $ is continuous for $ t \ge 0 $, the Laplace transform of $ f $ is…

In Exercises $86-89$ , the Laplace transform of a fwection $f(x)$ is the fun…

Use properties of the Laplace transform and the table of Laplace transforms …

A powerful tool in solving problems in engineering and physics is the Laplac…

Suppose that $ 0 \le f(t) \le Me^{at} $ and $ 0 \le f^\prime (t) \le Ke^{at}…

04:09

For the following exercises, determine whether the statement is true or fals…

02:02

Evaluate the integrals. If the integral diverges, answer "diverges.&quo…

01:25

For the following exercises, evaluate the definite integrals. Express answer…

00:58

For each of the following sequences, whose $n$th terms are indicated, state …

02:29

For the following problems, determine how parameter $a$ affects the solution…

02:31

Use substitution to convert the integrals to integrals of rational functions…

00:30

Compute the following integrals using the guidelines for integrating powers …

What are the integrating factors for the following differential equations?

01:13

Match the direction field with the given differential equations. Explain you…

09:17

Sketch the region bounded above by the curve, the $x$ -axis, and $x=1,$ and …

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.