Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 20


$$\frac{1}{2} \tan x \sec x+\frac{1}{2}\ln |\sec x+\tan x|+c$$

Work Step by Step

Since $$\int \sec ^{m} x d x=\frac{\tan x \sec ^{m-2} x}{m-1}+\frac{m-2}{m-1} \int \sec ^{m-2} x d x$$ Then \begin{align*} \int \sec ^{3} x d x&=\frac{1}{2} \tan x \sec x+\frac{1}{2} \int \sec x d x\\ &=\frac{1}{2} \tan x \sec x+\frac{1}{2}\ln |\sec x+\tan x|+c \end{align*}
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