Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 7 - Integration - 7.2 Substitution - 7.2 Exercises: 26

Answer

\[ - \frac{1}{{2\,{{\left( {x + 5} \right)}^4}}} + \frac{2}{{\,{{\left( {x + 5} \right)}^5}}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{2x}}{{\,{{\left( {x + 5} \right)}^6}}}dx} \hfill \\ Let\,u = x + 5\,\,,\,\,then\,\,x = u - 5 \hfill \\ So\,\,that \hfill \\ dx = du \hfill \\ Then \hfill \\ \int_{}^{} {\frac{{2\,\left( {u - 5} \right)}}{{{u^6}}}du\,} \hfill \\ Write\,\,the\,\,integrand\,\,as \hfill \\ 2\int_{}^{} {\,\left( {\frac{u}{{{u^6}}} - \frac{5}{{{u^6}}}} \right)du} \hfill \\ 2\int_{}^{} {\,\left( {{u^{ - 5}} - 5{u^{ - 6}}} \right)du} \hfill \\ Integrating \hfill \\ 2\,\left( {\frac{{{u^{ - 4}}}}{{ - 4}}} \right) - 10\,\left( {\frac{{{u^{ - 5}}}}{{ - 5}}} \right) + C \hfill \\ - \frac{1}{{2{u^4}}} + \frac{2}{{{u^5}}} + C \hfill \\ Substituting\,\,u = x + 5\,\,gives \hfill \\ - \frac{1}{{2\,{{\left( {x + 5} \right)}^4}}} + \frac{2}{{\,{{\left( {x + 5} \right)}^5}}} + C \hfill \\ \end{gathered} \]
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