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: 66

Answer

=1

Work Step by Step

\[\begin{gathered} \int_0^\infty {x{e^{ - x}}dx} \hfill \\ \hfill \\ use\,\,the\,\,Formula\,\,for\,\,in\,tegration\,\,by\,\,parts \hfill \\ \hfill \\ \int_{}^{} {udv = uv - \int_{}^{} {vdu} } \hfill \\ \hfill \\ set \hfill \\ du = x\,\,\,\,\,\,\,\,\,\,\,\,\,\,then\,\,\,du = dx \hfill \\ dv = {e^{ - x}}dx\,\,\,then\,\,\,\,v = {e^{ - x}} \hfill \\ \hfill \\ \operatorname{int} egrating \hfill \\ \hfill \\ \int_0^\infty {x{e^{ - x}}dx} \,\, = \,\,\,\mathop {\lim }\limits_{b \to \infty } \int_0^b {x{e^{ - x}}dx} \hfill \\ \hfill \\ = \mathop {\lim }\limits_{b \to \infty } \,\,\left[ {x\,\left( { - {e^{ - x}}} \right) - {e^{ - x}}} \right]_0^b \hfill \\ \hfill \\ use\,\,the\,\,ftc \hfill \\ \hfill \\ = \mathop {\lim }\limits_{b \to \infty } \,\,\left[ { - b\,\left( {{e^{ - b}}} \right) - {e^{ - b}} + 1} \right] \hfill \\ \hfill \\ evaluate\,\,the\,\,{\text{limit}} \hfill \\ \hfill \\ = 1 \hfill \\ \end{gathered} \]
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