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 - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 463: 61


$$\frac{1}{4} \cos \theta-\frac{1}{20} \cos 5 \theta+C $$

Work Step by Step

We integrate as follows: \begin{align*} \int \sin \theta \cos \theta \cos 3 \theta d \theta&=\frac{1}{2} \int 2 \sin \theta \cos \theta \cos 3 \theta d \theta\\ &=\frac{1}{2} \int \sin 2 \theta \cos 3 \theta d \theta\\ &=\frac{1}{2} \int \frac{1}{2}(\sin (2-3) \theta+\sin (2+3) \theta) d \theta\\ &=\frac{1}{4} \int(\sin (-\theta)+\sin 5 \theta) d \theta\\ &=\frac{1}{4} \int(-\sin \theta+\sin 5 \theta) d \theta\\ &=\frac{1}{4} \cos \theta-\frac{1}{20} \cos 5 \theta+C \end{align*}
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