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: 29

Answer

\[\frac{{\,{{\left( {1 + 3\ln x} \right)}^3}}}{9} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{\,{{\left( {1 + 3\ln x} \right)}^2}}}{x}dx} \hfill \\ Let\,\,u = 1 + 3\ln x\,\,,\,\,So\,\,that \hfill \\ du = \frac{3}{x}dx \hfill \\ Then \hfill \\ \frac{1}{3}\int_{}^{} {\frac{{3\,{{\left( {1 + 3\ln x} \right)}^2}}}{x}dx = \frac{1}{3}\int_{}^{} {{u^2}du} } \hfill \\ Integrating \hfill \\ \frac{1}{3}\,\left( {\frac{{{u^3}}}{3}} \right) + C \hfill \\ \frac{{{u^3}}}{9} + C \hfill \\ Substituting\,\,u = 1 + 3\ln x\,\,,\,\,gives \hfill \\ \frac{{\,{{\left( {1 + 3\ln x} \right)}^3}}}{9} + C \hfill \\ \end{gathered} \]
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