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: 31

Answer

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

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{{e^{2x}}}}{{{e^{2x}} + 5}}} dx \hfill \\ Let\,\,u = {e^{2x}} + 5\,\,,\,\,So\,\,that \hfill \\ du = 2{e^{2x}}dx \hfill \\ Then \hfill \\ \frac{1}{2}\int_{}^{} {\frac{{2{e^{2x}}}}{{{e^{2x}} + 5}}dx} = \frac{1}{2}\int_{}^{} {\frac{{du}}{u}} \hfill \\ Integrating \hfill \\ \frac{1}{2}\ln \left| u \right| + C \hfill \\ Substituting\,\,u = {e^{2x}} + 5\,\,gives \hfill \\ \frac{1}{2}\ln \,\left( {{e^{2x}} + 5} \right) + C \hfill \\ \end{gathered} \]
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