Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.2 Integration by Parts - 7.2 Exercises - Page 520: 24

Answer

$$\frac{3}{{13}}{e^{3x}}\cos 2x + \frac{2}{{13}}{e^{3x}}\sin 2x + C$$

Work Step by Step

$$\eqalign{ & \int {{e^{3x}}\cos 2x} dx \cr & {\text{substitute }}u = \cos 2x,{\text{ }}du = - 2\sin 2xdx \cr & dv = {e^{3x}}dx,{\text{ }}v = \frac{1}{3}{e^{3x}} \cr & {\text{applying integration by parts}}{\text{, we have}} \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x - \int {\frac{1}{3}{e^{3x}}\left( { - 2\sin 2xdx} \right)} \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{3}\int {{e^{3x}}\sin 2xdx} \cr & {\text{substitute }}u = \sin 2x,{\text{ }}du = 2\cos 2xdx \cr & dv = {e^{3x}}dx,{\text{ }}v = \frac{1}{3}{e^{3x}} \cr & {\text{applying integration by parts}}{\text{, we have}} \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{3}\left( {\frac{1}{3}{e^{3x}}\sin 2x - \int {\frac{1}{3}{e^{3x}}\left( {2\cos 2xdx} \right)} } \right) \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{3}\left( {\frac{1}{3}{e^{3x}}\sin 2x - \frac{2}{3}\int {{e^{3x}}\cos 2xdx} } \right) \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x + \left( {\frac{2}{9}{e^{3x}}\sin 2x - \frac{4}{9}\int {{e^{3x}}\cos 2xdx} } \right) \cr & \int {{e^{3x}}\cos 2x} dx = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{9}{e^{3x}}\sin 2x - \frac{4}{9}\int {{e^{3x}}\cos 2xdx} \cr & {\text{solving for }}\int {{e^{3x}}\cos 2x} dx \cr & \int {{e^{3x}}\cos 2x} dx + \frac{4}{9}\int {{e^{3x}}\cos 2xdx} = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{9}{e^{3x}}\sin 2x \cr & \frac{{13}}{9}\int {{e^{3x}}\cos 2xdx} = \frac{1}{3}{e^{3x}}\cos 2x + \frac{2}{9}{e^{3x}}\sin 2x \cr & \int {{e^{3x}}\cos 2xdx} = \left( {\frac{9}{{13}}} \right)\frac{1}{3}{e^{3x}}\cos 2x + \left( {\frac{9}{{13}}} \right)\frac{2}{9}{e^{3x}}\sin 2x + C \cr & \int {{e^{3x}}\cos 2xdx} = \frac{3}{{13}}{e^{3x}}\cos 2x + \frac{2}{{13}}{e^{3x}}\sin 2x + C \cr} $$
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