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: 34


$$-\frac{1}{2} \cot x \csc x+\frac{1}{2} \ln |\csc x-\cot x|+c$$

Work Step by Step

Given $$\int \csc ^{3} x d x $$ Use $$\int \csc ^{m} x d x=-\frac{\cot x \csc ^{m-2} x}{m-1}+\frac{m-2}{m-1} \int \csc ^{m-2} x d x $$ Then \begin{aligned} \int \csc ^{3} x d x &=-\frac{\cot x \csc x}{2}+\frac{1}{2} \int \csc x d x \\ &=-\frac{1}{2} \cot x \csc x+\frac{1}{2} \int \csc x d x \\ &=-\frac{1}{2} \cot x \csc x+\frac{1}{2} \ln |\csc x-\cot x|+c \end{aligned}
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