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 403: 31


$$\frac{-1}{2}\sin^{-2}x -2\ln \sin x +\frac{1}{2}\sin^2 x+c$$

Work Step by Step

\begin{align*} \int \frac{\cos ^{5} x}{\sin ^{3} x} d x&= \int \sin^{-3}x\cos^4x\cos xdx\\ &= \int \sin^{-3}x(1-\sin^2x)^2\cos xdx\\ &=\int \sin^{-3}x(1-2\sin^2 x+\sin^4x)\cos xdx\\ &= \int \left( \sin^{-3}x -2\frac{1}{\sin x} + \sin x\right)\cos xdx\\ &= \frac{-1}{2}\sin^{-2}x -2\ln \sin x +\frac{1}{2}\sin^2 x+c \end{align*}
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