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Show that if $ 0 \le f(t) \le Me^{at} $ for $ t \ge 0 $, where $ M $ and $ a $ are constants, then the Laplace transform $ F(s) $ exists for $ s > a $.

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Hence $\int_{0}^{+\infty} M e^{(a-s) t} d t$ is convergent and then by the Comparison Theoremwe have that $\int_{0}^{+\infty} f(t) e^{-s t} d t$ is convergent and therefore the Laplace transform$F(s)$ exists.

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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