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 - Practice Exercises - Page 518: 42


$$\sec \theta+\cos \theta+C $$

Work Step by Step

We evaluate the trigonometric integral as follows: \begin{align*} \int \sec ^{2} \theta \sin ^{3} \theta d \theta&=\int \frac{\sin \theta\left(1-\cos ^{2} \theta\right)}{\cos ^{2} \theta}\\ &=\int \frac{\sin \theta}{\cos ^{2} \theta} d \theta-\int \sin \theta d \theta\\ &=\cos ^{-1} \theta-(-\cos \theta)+C\\ &=\sec \theta+\cos \theta+C \end{align*} Where we used the fact that $\sin^2 x + \cos^2 x=1$
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