Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 23


$$-\cos \theta +\frac{2}{3} \cos^3 \theta-\frac{1}{5} \cos^5 \theta+C$$

Work Step by Step

\begin{aligned} \int \sin ^{5} \theta d \theta &=\int \sin ^{4} \theta \sin \theta d \theta \\ &=\int\left(1-\cos ^{2} \theta\right)^{2} \sin \theta d \theta \\ &=\int\left(1-2 \cos ^{2} \theta+\cos ^{4} \theta\right) \sin \theta d \theta \\ &=\int \sin \theta d \theta-2 \int \cos ^{2} \theta \sin \theta d \theta+\int \cos ^{4} \theta \sin \theta d \theta \\ &=-\cos \theta +\frac{2}{3} \cos^3 \theta-\frac{1}{5} \cos^5 \theta+C \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.