Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.6 Other Integration Strategies - 7.6 Exercises: 20

Answer

\[ = \frac{1}{{16}}\ln \,\left( {\frac{{{v^2}}}{{{v^2} + 8}}} \right) + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{dv}}{{v\,\left( {{v^2} + 8} \right)}}} \hfill \\ \hfill \\ using\,\,formula\,\,85 \hfill \\ \hfill \\ \int_{}^{} {\frac{{dx}}{{x\,\left( {{a^2} + {x^2}} \right)}}} = \frac{1}{{2{a^2}}}\ln \,\left( {\frac{{{x^2}}}{{{a^2} + {x^2}}}} \right) + C \hfill \\ \hfill \\ with\,\,\,{a^2} = 8 \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \int_{}^{} {\frac{{dv}}{{v\,\left( {{v^2} + 8} \right)}}} = \,\frac{1}{{2\,{{\left( 8 \right)}^2}}}\ln \,\left( {\frac{{{v^2}}}{{{v^2} + 8}}} \right) + C \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \frac{1}{{16}}\ln \,\left( {\frac{{{v^2}}}{{{v^2} + 8}}} \right) + C \hfill \\ \end{gathered} \]
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