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: 71


$$\int \tan ^{k} x d x =\frac{\tan^{k-1}}{k-1}- \int \tan ^{k-2} x d x $$

Work Step by Step

\begin{aligned} \int \tan ^{k} x d x &=\int \tan ^{k-2} x \tan ^{2} x d x \\ &=\int \tan ^{k-2} x\left(\sec ^{2} x-1\right) d x \\ &=\int \tan ^{k-2} x \sec ^{2} x d x-\int \tan ^{k-2} x d x \\ &=\frac{\tan^{k-1}}{k-1}- \int \tan ^{k-2} x d x \end{aligned}
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