Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 396: 59


$$\frac{1}{3} \cos ^{2} x \sin x+\frac{2}{3} \sin x+C$$

Work Step by Step

Given $$ \int \cos ^{3} x d x$$ Use $$\int \cos ^{n} x d x = \frac{1}{n} \cos^{n-1} x\sin x+\frac{(n-1)}{n} \int \cos^{n-2}x dx\\ $$ Then for $n=3$ \begin{aligned} \int \cos ^{3} x d x &=\frac{1}{3} \cos ^{2} x \sin x+\frac{2}{3} \int \cos x d x \\ &=\frac{1}{3} \cos ^{2} x \sin x+\frac{2}{3} \sin x+C \end{aligned}
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