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 8.6 - Integral Tables and Computer Algebra Systems - Exercises 8.6 - Page 481: 46


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

Work Step by Step

Use the Reduction Formula: $\int\cot^naxdx=-\dfrac{\cot^{n-1}ax}{a(n-1)}-\int\cot^{n-2} (ax) \space dx$ Let $I=8[-\dfrac{\cot^3t}{3}-\int\cot^2tdt ] \\=8[-\dfrac{\cot^3t}{3}-(-\dfrac{\cot t}{1}-\int\cot^0tdt\Big)]\\=8[-\dfrac{\cot^3t}{3}+\cot t+t]+C\\=-\dfrac{8}{3}\cot^3t+8\cot t+8t+C$
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