Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Practice Exercises - Page 519: 90

Answer

$$\frac{{{{\tan }^2}t}}{2} + \ln \left| {\cos t} \right| + C$$

Work Step by Step

$$\eqalign{ & \int {{{\tan }^3}t} dt \cr & {\text{Write the integrand as }}{\tan ^{2 + 1}}t \cr & = \int {{{\tan }^{2 + 1}}t} dt \cr & {\text{use the property }}{a^{m + }}^n = {a^m}{a^n} \cr & = \int {{{\tan }^2}t\tan t} dt \cr & {\text{use the trigonometric identity ta}}{{\text{n}}^2}x = {\sec ^2}x - 1 \cr & = \int {\left( {{{\sec }^2}t - 1} \right)\tan t} dt \cr & {\text{distribute}} \cr & = \int {{{\sec }^2}t\tan t} dt - \int {\tan t} dt \cr & {\text{integrate by using basic rules}} \cr & = \frac{{{{\tan }^2}t}}{2} - \left( { - \ln \left| {\cos t} \right|} \right) + C \cr & = \frac{{{{\tan }^2}t}}{2} + \ln \left| {\cos t} \right| + C \cr} $$
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