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: 83

Answer

\[ = \frac{a}{{{a^2} + {b^2}}}\,\,and\,\frac{b}{{{a^2} + {b^2}}}\]

Work Step by Step

\[\begin{gathered} part\,a \hfill \\ \hfill \\ we\,\,\,know\,\,the\,\,formula \hfill \\ \hfill \\ \int_{}^{} {{e^{ - ax}}\,\cos \,\,bx\,dx = \frac{{{e^{ - ax}}}}{{{a^2} + {b^2}}}} \,\,\left[ { - a\cos bx + b\sin bx} \right] \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \int_0^\infty {{e^{ - ax}}\,\cos \,\,bx\,dx = \mathop {\lim }\limits_{c \to \infty } \frac{{{e^{ - ax}}}}{{{a^2} + {b^2}}}} \,\,\left[ { - a\cos bx + b\sin bx} \right]_0^c \hfill \\ \hfill \\ use\,\,the\,\,ftc \hfill \\ \hfill \\ = \mathop {\lim }\limits_{c \to \infty } \frac{{{e^{ - ac}}}}{{{a^2} + {b^2}}}\,\,\left[ { - a\cos bc + b\sin bc} \right] + \frac{a}{{{a^2} + {b^2}}} \hfill \\ \hfill \\ {\text{evaluate}}\,\,{\text{the}}\,\,{\text{limit}} \hfill \\ \hfill \\ = \frac{a}{{{a^2} + {b^2}}} \hfill \\ \hfill \\ part\,\,b \hfill \\ \hfill \\ we\,\,\,know\,\,the\,\,formula \hfill \\ \hfill \\ \int_{}^{} {{e^{ - ax}}\,\sin \,bx\,dx = } \frac{{{e^{ - ax}}}}{{{a^2} + {b^2}}}\,\,\left[ { - a\sin bx - b\cos bx} \right] \hfill \\ \hfill \\ then \hfill \\ \hfill \\ = \mathop {\lim }\limits_{c \to \infty } \frac{{{e^{ - ax}}}}{{{a^2} + {b^2}}}\,\,\left[ { - a\sin bx - b\cos bx} \right]_0^c \hfill \\ \hfill \\ use\,\,the\,\,ftc \hfill \\ \hfill \\ = \mathop {\lim }\limits_{c \to \infty } \frac{{{e^{ - ac}}}}{{{a^2} + {b^2}}}\,\,\left[ { - a\sin bx - b\cos bx} \right] + \frac{b}{{{a^2} + {b^2}}} \hfill \\ \hfill \\ {\text{evaluate}}\,\,{\text{the}}\,\,{\text{limit}} \hfill \\ \hfill \\ = \frac{b}{{{a^2} + {b^2}}} \hfill \\ \end{gathered} \]
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