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: 63

Answer

$\text{(a) }2\\\text{(b) }0$

Work Step by Step

\[\begin{gathered} part\,\,a \hfill \\ \int_{ - \infty }^\infty {{e^{ - \left| x \right|}}} dx \hfill \\ \hfill \\ the\,\,{\text{ }}function\,\,{e^{ - \left| x \right|}}is\,\,even,{\text{ then}} \hfill \\ \hfill \\ \int_{ - \infty }^\infty {{e^{ - \left| x \right|}}} dx = \hfill \\ \hfill \\ can\,\,be\,\,\,write\,\,as \hfill \\ \hfill \\ 2\int_0^\infty {{e^{ - x}}} dx = 2 \hfill \\ \hfill \\ part\,b \hfill \\ \hfill \\ \int_{ - \infty }^\infty {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx \hfill \\ \hfill \\ {\text{apply}}\,\,\,{\text{the}}\,\,{\text{properties}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{integral}} \hfill \\ \hfill \\ \int_{ - \infty }^\infty {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx = \int_0^\infty {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx + \int_{ - \infty }^0 {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx \hfill \\ \hfill \\ the\,\,fuction\,\,is\,\,odd,\,\,then \hfill \\ \hfill \\ = \int_0^\infty {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx - \int_0^\infty {\frac{{{x^3}}}{{1 + {x^8}}}\,} dx = 0 \hfill \\ \end{gathered} \]
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