Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Additional and Advanced Exercises - Page 521: 33

Answer

$$\frac{{{e^{2x}}}}{{13}}\left( {3\sin 3x + 2\cos 3x} \right) + C$$

Work Step by Step

$\begin{gathered} \int {{e^{2x}}\cos 3x} dx \hfill \\ {\text{Use tabular integration to evaluate the integral}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {{e^{2x}}{\text{ and its derivatives}}}&{}&{\cos 3x{\text{ and its integrals}}} \\ {{e^{2x}}}&{\mathop {\left( + \right)}\limits_ \searrow }&{\cos 3x} \\ {2{e^{2x}}}&{\mathop {\left( - \right)}\limits_ \searrow }&{\frac{1}{3}\sin 3x} \\ {4{e^{2x}}}&{\left( + \right)}&{ - \frac{1}{9}\cos 3x} \end{array} \hfill \\ \end{gathered} $ $$\eqalign{ & {\text{We need to stop here because it is the same as the first row except }} \cr & {\text{for constants }}\left( {4{\text{ on the left and }} - \frac{1}{9}{\text{ on the right}}} \right).{\text{ Then}}{\text{,}} \cr & {\text{We interpret the table as saying }} \cr & \int {{e^{2x}}\cos 3x} dx = \left( {{e^{2x}}} \right)\left( {\frac{1}{3}\sin 3x} \right) + \left( {2{e^{2x}}} \right)\left( { + \frac{1}{9}\cos 3x} \right) + \int {\left( {4{e^{2x}}} \right)\left( { - \frac{1}{9}\cos 3x} \right)} dx \cr & \int {{e^{2x}}\cos 3x} dx = \frac{1}{3}{e^{2x}}\sin 3x + \frac{2}{9}{e^{2x}}\cos 3x - \frac{4}{9}\int {{e^{2x}}\cos 3x} dx \cr & \cr & {\text{Solve for }}\int {{e^{2x}}\cos 3x} dx \cr & \int {{e^{2x}}\cos 3x} dx + \frac{4}{9}\int {{e^{2x}}\cos 3x} dx = \frac{1}{3}{e^{2x}}\sin 3x + \frac{2}{9}{e^{2x}}\cos 3x \cr & \frac{{13}}{9}\int {{e^{2x}}\cos 3x} dx = \frac{1}{3}{e^{2x}}\sin 3x + \frac{2}{9}{e^{2x}}\cos 3x + C \cr & \int {{e^{2x}}\cos 3x} dx = \frac{9}{{13}}\left( {\frac{1}{3}{e^{2x}}\sin 3x + \frac{2}{9}{e^{2x}}\cos 3x} \right) + C \cr & \int {{e^{2x}}\cos 3x} dx = \frac{3}{{13}}{e^{2x}}\sin 3x + \frac{2}{{13}}{e^{2x}}\cos 3x + C \cr & {\text{Factoring gives}} \cr & \int {{e^{2x}}\cos 3x} dx = \frac{{{e^{2x}}}}{{13}}\left( {3\sin 3x + 2\cos 3x} \right) + C \cr} $$
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