University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 8 - Section 8.5 - Integral Tables and Computer Algebra Systems - Exercises - Page 451: 46


$$\int 8\cot^4tdt=-\frac{8}{3}\cot^3t+8\cot t+8t+C$$

Work Step by Step

$$A=\int 8\cot^4tdt$$ Use Reduction Formula 94, which states that $$\int\cot^naxdx=-\frac{\cot^{n-1}ax}{a(n-1)}-\int\cot^{n-2}axdx$$ for $n=4$ and $a=1$ $$A=8\Big[-\frac{\cot^3t}{3}-\int\cot^2tdt\Big]$$ Use Formula 94 again, this time for $n=2$ and $a=1$ $$A=8\Big[-\frac{\cot^3t}{3}-\Big(-\frac{\cot t}{1}-\int\cot^0tdt\Big)\Big]$$ $$A=8\Big[-\frac{\cot^3t}{3}-\Big(-\cot t-\int 1dt\Big)\Big]$$ $$A=8\Big[-\frac{\cot^3t}{3}+\cot t+t\Big]+C$$ $$A=-\frac{8}{3}\cot^3t+8\cot t+8t+C$$
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