Calculus 8th Edition

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

Chapter 7 - Techniques of Integration - 7.5 Strategy for Integration - 7.5 Exercises - Page 548: 80

Answer

\[\ln |\sin x\cos x+1|+C\] Where $C$ is constant of integration

Work Step by Step

Let \[I=\int\frac{\sec x\,\cos 2x}{\sin x+\sec x}\:dx\] \[I=\int\frac{\sec x\,\cos 2x}{\sin x+\frac{1}{\cos x}}\:dx\] \[I=\int\frac{\cos 2x}{\sin x\:\cos x+1}\:dx\] Put $t=\sin x\:\cos x+1\;\;\;...(1)$ \[\Rightarrow dt=(\cos^2 x-\sin^2 x)dx=\cos 2x \:dx\] \[I=\int\frac{dt}{t}=\ln|t|+C\] Where $C$ is constant of integration Using (1) \[I=\ln |\sin x\cos x+1|+C\] Hence, \[\int\frac{\sec x\,\cos 2x}{\sin x+\sec x}=\ln |\sin x\cos x+1|+C\]
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