Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.2 Trigonometric Integrals - 7.2 Exercises - Page 524: 27


$\frac{1}{3}\sec^3 x-sec(x)+C$

Work Step by Step

$\int\tan^3 x\sec x\ dx$ Since the power of tangent is odd, save a factor of $\sec x\tan x$ and express the remaining factors in terms of $\sec x$: $=\int\tan^2 x\sec x\tan x\ dx$ $=\int(\sec^2 x-1)\sec x\tan x\ dx$ Let $u=\sec x$. Then $du=\sec x\tan x\ dx$. $=\int(u^2-1)du$ $=\frac{1}{3}u^3-u+C$ $=\boxed{\frac{1}{3}\sec^3 x-sec (x)+C}$
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