## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 8: Techniques of Integration - Practice Exercises - Page 519: 114

#### Answer

$$\frac{1}{2}x - \frac{1}{2}\ln \left| {\sin x + \cos x} \right| + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{{\sin x}}{{\sin x + \cos x}}} dx \cr & {\text{Add and subtract }}\cos x{\text{ into the numerator}} \cr & = \int {\frac{{\sin x + \cos x - \cos x}}{{\sin x + \cos x}}} dx \cr & {\text{distribute}} \cr & = \int {\left( {\frac{{\sin x + \cos x}}{{\sin x + \cos x}} - \frac{{\cos x}}{{\sin x + \cos x}}} \right)} dx \cr & = \int {\left( {1 - \frac{{\cos x}}{{\sin x + \cos x}}} \right)} dx \cr & = \int {dx} - \int {\frac{{\cos x}}{{\sin x + \cos x}}} dx \cr & = \int {dx} - \frac{1}{2}\int {\frac{{\cos x + \cos x}}{{\sin x + \cos x}}} dx \cr & {\text{Add and subtract }}\sin x{\text{ into the numerator}} \cr & = \int {dx} - \frac{1}{2}\int {\frac{{\cos x + \cos x}}{{\sin x + \cos x}}} dx \cr & = \int {dx} - \frac{1}{2}\int {\frac{{\cos x - \sin x + \sin x + \cos x}}{{\sin x + \cos x}}} dx \cr & = \int {dx} - \frac{1}{2}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx - \frac{1}{2}\int {\frac{{\sin x + \cos x}}{{\sin x + \cos x}}} dx \cr & = \frac{1}{2}\int {dx} - \frac{1}{2}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx \cr & = \frac{1}{2}x - \frac{1}{2}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx \cr & \cr & {\text{Integrate }}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx{\text{ by the substitution method}} \cr & \,\,\,u = \sin x + \cos x,\,\,\,du = \left( {\cos x - \sin x} \right)dx \cr & \,\, - \frac{1}{2}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx = - \frac{1}{2}\int {\frac{{du}}{u}} = \ln \left| u \right| + C \cr & \,\,\,{\text{substitute }}\sin x + \cos x{\text{ for }}u \cr & \,\,\,\, = - \frac{1}{2}\int {\frac{{du}}{u}} = - \frac{1}{2}\ln \left| {\sin x + \cos x} \right| + C \cr & \cr & {\text{Then}} \cr & = \frac{1}{2}x - \frac{1}{2}\int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}} dx = \frac{1}{2}x - \frac{1}{2}\ln \left| {\sin x + \cos x} \right| + C \cr}

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