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 - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 462: 50


$$-\frac{8}{3} \cot ^{3} t+8 \cot t+8 t+C $$

Work Step by Step

We integrate as follows: \begin{align*} \int 8 \cot ^{4} t d t&=8 \int\left(\csc ^{2} t-1\right) \cot ^{2} t d t\\ &=8 \int \csc ^{2} t \cot ^{2} t d t-8 \int \cot ^{2} t d t\\ &=-\frac{8}{3} \cot ^{3} t-8 \int\left(\csc ^{2} t-1\right) d t\\ &=-\frac{8}{3} \cot ^{3} t+8 \cot t+8 t+C \end{align*}
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