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Suppose that $ 0 \le f(t) \le Me^{at} $ and $ 0 \le f^\prime (t) \le Ke^{at} $ for $ t \ge 0 $, where $ f^\prime $ is continuous. If the Laplace transform of $ f(t) $ is $ F(s) $ and the Laplace transform of $ f^\prime (t) $ is $ G(s) $ , show that$ G(s) = s F(s) - f(0) $ $ s > a $

$\lim _{t \rightarrow \infty} f(t) e^{-s t}=0$ for $s>a \Rightarrow G(s)=0-f(0)+s F(s)=s F(s)-f(0)$ for $s>a$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

Lectures

01:11

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04:13

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