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: 10

Answer

\[\frac{1}{{15}}\,{\left( {5{r^2} + 2} \right)^{3/2}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} r \sqrt {5{r^2} + 2} dr \hfill \\ Let\,\,\,u = 5{r^2} + 2 \hfill \\ So\,\,\,that\,\,du = 10rdr\,\,\,Then \hfill \\ \int_{}^{} r \sqrt {5{r^2} + 2} dr = \frac{1}{{10}}\int_{}^{} {10r\sqrt {5{r^2} + 2} dr} \hfill \\ = \frac{1}{{10}}\int_{}^{} {\sqrt u du} \hfill \\ \frac{1}{{10}}\int_{}^{} {{u^{1/2}}du} \hfill \\ Power\,\,rule \hfill \\ \frac{1}{{10}}\,\left( {\frac{{{u^{3/2}}}}{{3/2}}} \right) + C \hfill \\ \frac{1}{{15}}{u^{3/2}} + C \hfill \\ Substituting\,\,5{r^2} + 2\,\,for\,\,u\,\,gives \hfill \\ \frac{1}{{15}}\,{\left( {5{r^2} + 2} \right)^{3/2}} + C \hfill \\ \end{gathered} \]
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