University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Section 8.2 - Trigonometric Integrals - Exercises - Page 434: 50



Work Step by Step

$$A=\int\cot^3t\csc^4tdt=\int\cot^3t\csc^2t(\csc^2tdt)$$ $$A=\int\cot^3t(\cot^2t+1)(\csc^2tdt)$$ $$A=-\int\cot^3t(\cot^2t+1)d(\cot t)$$ We set $u=\cot t$ $$A=-\int u^3(u^2+1)du$$ $$A=-\int(u^5+u^3)du$$ $$A=-\frac{u^6}{6}-\frac{u^4}{4}+C$$ $$A=-\frac{\cot^6t}{6}-\frac{\cot^4t}{4}+C$$
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