University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Section 8.2 - Trigonometric Integrals - Exercises - Page 435: 63


$$\int\frac{\sec^3x}{\tan x}dx=\sec x-\ln|\csc x+\cot x|+C$$

Work Step by Step

$$A=\int\frac{\sec^3x}{\tan x}dx=\int\frac{\sec^2x\sec x}{\tan x}dx$$ $$A=\int\frac{(\tan^2x+1)\sec x}{\tan x}dx$$ $$A=\int\frac{\tan^2x\sec x}{\tan x}dx+\int\frac{\sec x}{\tan x}dx$$ $$A=\int\sec x\tan xdx+\int\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}dx$$ $$A=\int d(\sec x)+\int\frac{1}{\sin x}dx$$ $$A=\sec x+\int\csc xdx$$ $$A=\sec x-\ln|\csc x+\cot x|+C$$
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