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 - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 463: 63


$$\sec x-\ln |\csc x+\cot x|+C $$

Work Step by Step

We integrate as follows: \begin{align*} \int \frac{\sec ^{3} x}{\tan x} d x&=\int \frac{\sec ^{2} x \sec x}{\tan x} d x\\ &=\int \frac{\left(\tan ^{2} x+1\right) \sec x}{\tan x} d x\\ &=\int \frac{\tan ^{2} x \sec x}{\tan x} d x+\int \frac{\sec x}{\tan x} d x\\ &=\int \tan x \sec x d x+\int \csc x d x \\ &=\sec x-\ln |\csc x+\cot x|+C \end{align*}
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