University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 7 - Section 7.1 - The Logarithm Defined as an Integral - Exercises - Page 401: 8


$2 \sqrt {\ln (\sec x +\tan x)}+c$

Work Step by Step

Solve $\int\dfrac{\sec xdx}{\sqrt {\ln (\sec x +\tan x)}}$ Let $p=\ln (\sec x +\tan x)$ and $dp=\sec xdx$ This implies $\int\dfrac{\sec xdx}{\sqrt {\ln (\sec x +\tan x)}}=\int\dfrac{dp}{\sqrt p}=2p^{1/2}+c=2 \sqrt p+c$ Since, $p=\ln (\sec x +\tan x)$ Thus, $\int\dfrac{\sec xdx}{\sqrt {\ln (\sec x +\tan x)}}=2 \sqrt {\ln (\sec x +\tan x)}+c$
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