Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 14

Answer

\[ = \frac{1}{{20}}\sin 20x - \frac{1}{{20}}\,\left( {\frac{{\sin 20x}}{3}} \right) + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\cos }^3}20xdx} \hfill \\ \hfill \\ rewrite \hfill \\ \hfill \\ = \int_{}^{} {{{\cos }^2}20x \cdot \cos 20xdx} \hfill \\ \hfill \\ where \hfill \\ \hfill \\ {\cos ^2}20x = 1 - {\sin ^2}20x \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {1 - {{\sin }^2}20x} \right)\cos 20xdx} \hfill \\ \hfill \\ distribute \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\cos 20x} \right)dx} - \int_{}^{} {{{\sin }^2}20x\cos 20xdx} \hfill \\ \hfill \\ rewriting \hfill \\ \hfill \\ = \frac{1}{{20}}\int_{}^{} {\cos 20x\,\left( {20} \right)dx} - \frac{1}{{20}}\int_{}^{} {{{\sin }^2}20x\cos 20x\,\left( {20} \right)dx} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \frac{1}{{20}}\sin 20x - \frac{1}{{20}}\,\left( {\frac{{\sin 20x}}{3}} \right) + C \hfill \\ \end{gathered} \]
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