Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.5 - Strategy for Integration - 7.5 Exercises - Page 508: 62


$$\int \frac{1}{1+cos^{2}\theta}d\theta=\frac{1}{\sqrt{2}}arctan(\frac{tan\theta}{\sqrt{2}})+C$$

Work Step by Step

$$\int \frac{1}{1+cos^{2}\theta}d\theta=\int \frac{1}{1+cos^{2}\theta}\frac{sec^{2}\theta}{sec^{2}\theta}d\theta$$ $$=\int\frac{sec^{2}\theta}{sec^{2}\theta+1}d\theta=\int \frac{sec^{2}\theta}{tan^{2}\theta+2}d\theta$$ $$=\int \frac{1}{tan^{2}\theta+2}d(tan\theta)$$ $$=\frac{1}{\sqrt{2}}arctan(\frac{tan\theta}{\sqrt{2}})+C$$
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