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: 23

Answer

\[\frac{{\,{{\left( {p + 1} \right)}^7}}}{7} - \frac{{\,{{\left( {p + 1} \right)}^6}}}{6} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {p\,{{\left( {p + 1} \right)}^5}dp} \hfill \\ \hfill \\ Let\,\,u = p + 1\,\,,\,\,then\,\,p = u - 1 \hfill \\ So\,\,that\,\,dp = du \hfill \\ \int_{}^{} {\,\left( {u - 1} \right){u^5}du} \hfill \\ \int_{}^{} {\,\left( {{u^6} - {u^5}} \right)du} \hfill \\ Power\,\,rule \hfill \\ \frac{{{u^{6 + 1}}}}{{6 + 1}} - \frac{{{u^{5 + 1}}}}{{5 + 1}} + C \hfill \\ \frac{{{u^7}}}{7} - \frac{{{u^6}}}{6} + C \hfill \\ Substituting\,\,u = p + 1\,\,gives \hfill \\ \frac{{\,{{\left( {p + 1} \right)}^7}}}{7} - \frac{{\,{{\left( {p + 1} \right)}^6}}}{6} + C \hfill \\ \end{gathered} \]
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