Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.8 Improper Integrals - 7.8 Exercises: 91

Answer

\[ = \frac{1}{{s - a}}\]

Work Step by Step

\[\begin{gathered} f\,\left( t \right) = {e^{et}}\,\,\,\,\,\, \to \,\,\,\,\,f\,\left( s \right) = \frac{1}{{s - a}} \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \,\mathcal{L}\,\left[ {{e^{at}}u\,\left( t \right)} \right] = \int_0^\infty {{e^{at}}{e^{st}}\,dt} \hfill \\ \hfill \\ multiply\,\,and\,\,integrate \hfill \\ \hfill \\ = \left. {\frac{1}{{s - a}}\,\left( {s - a} \right)t} \right|_0^\infty \hfill \\ \hfill \\ use\,\,\,the\,\,ftc \hfill \\ \hfill \\ = \frac{1}{{s - a}} \hfill \\ \end{gathered} \]
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