Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.5 - Strategy for Integration - 7.5 Exercises - Page 508: 36


$$\int \frac{1+sinx}{1+cosx}dx=-cotx+ln\left |cscx -cotx\right |+cscx-ln\left | sinx \right |+C$$

Work Step by Step

$$\int \frac{1+sinx}{1+cosx}dx=\int \frac{1+sinx}{1+cosx}\frac{1-cosx}{1-cosx}dx$$ $$=\int \frac{1+sinx-cosx-sinx\,cosx}{sin^{2}x}dx$$ $$=\int (csc^{2}x+csc\,x-cscx\,cotx-cotx)dx$$ $$=-cotx+ln\left |cscx -cotx\right |+cscx-ln\left | sinx \right |+C$$
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