Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 404: 75


$$-\cos x(\ln \sin x) +\ln |\csc x-\cot x|+\cos x+C$$

Work Step by Step

Given $$\int \sin x \ln (\sin x) d x$$ Let \begin{align*} u&=\ln (\sin x)\ \ \ \ \ \ \ dv=\sin xdx\\ du&= \frac{\cos x}{\sin x}dx\ \ \ \ \ \ \ \ v=-\cos x \end{align*} Then \begin{align*} \int \sin x \ln (\sin x) d x&= -\cos x\ln (\sin x)+\int \frac{\cos^2 x}{\sin x}dx\\ &= -\cos x\ln (\sin x)+\int \frac{1-\sin^2 x}{\sin x}dx\\ &= -\cos x\ln (\sin x)+\int [\csc x-\sin x]dx\\ &= -\cos x\ln (\sin x) +\ln |\csc x-\cot x|+\cos x+C \end{align*}
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