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: 90

Answer

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

Work Step by Step

\[\begin{gathered} f\,\left( t \right) = 1\,\,\,\, \to \,\,f\,\left( s \right) = \frac{1}{s} \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \mathcal{L} = \int_0^\infty {{e^{ - st}}\,\left( 1 \right)dt} \hfill \\ \hfill \\ \mathcal{L} = \mathop {\lim }\limits_{b \to \infty } \int_0^b {{e^{ - st}}\,dt} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ \mathcal{L} = \left. {\mathop {\lim }\limits_{b \to \infty } \frac{{ - {e^{ - st}}}}{s}} \right|_0^b \hfill \\ \hfill \\ use\,\,the\,\,ftc \hfill \\ \hfill \\ = \mathop {\lim }\limits_{b \to \infty } \frac{{ - {e^{sb}} + 1}}{s} \hfill \\ \hfill \\ evaluate\,\,the\,\,limit \hfill \\ \hfill \\ = \frac{1}{s} \hfill \\ \end{gathered} \]
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