Calculus 8th Edition

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

Chapter 7 - Techniques of Integration - Review - Exercises - Page 578: 36

Answer

\[\ln|\cos\theta+\sin\theta|+C\]

Work Step by Step

Let \[I=\int\frac{1-\tan\theta}{1+\tan\theta}d\theta\] \[I=\int\frac{1-\left(\frac{\sin\theta}{\cos\theta}\right)}{1+\left(\frac{\sin\theta}{\cos\theta}\right)}d\theta\] \[I=\int\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}\;d\theta\;\;\;\ldots (1)\] Substitute $\; \cos\theta+\sin\theta=t\;\;\;\ldots (2)$ \[\Rightarrow (\cos\theta-\sin\theta)d\theta=dt\] (1) becomes \[I=\int\frac{dt}{t}=\ln|t|+C\] Where $C$ ia constant of integration From (2) \[I=\ln|\cos\theta+\sin\theta|+C\] Hence, \[I=\ln|\cos\theta+\sin\theta|+C\;\;.\]
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