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: 11

Answer

\[\frac{1}{2}{e^{2{x^3}}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {3{x^2}{e^{2{x^3}}}dx} \hfill \\ Let\,\,u = 2{x^3} \hfill \\ So\,\,that\,\,du = 6{x^2}dx.\,\,Then \hfill \\ \int_{}^{} {3{x^2}{e^{2{x^3}}}dx} = \frac{1}{2}\int_{}^{} {{e^{2{x^3}}}\,\left( {6{x^2}} \right)dx} \hfill \\ \frac{1}{2}\int_{}^{} {{e^u}du} \hfill \\ Integrating \hfill \\ \frac{1}{2}{e^u} + C \hfill \\ Substituting\,\,u = 2{x^3}\,\,for\,\,u\,\,gives \hfill \\ \frac{1}{2}{e^{2{x^3}}} + C \hfill \\ \end{gathered} \]
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