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: 30

Answer

\[\frac{2}{3}\,{\left( {2 + \ln x} \right)^{3/2}} + C\]

Work Step by Step

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