Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.2 Trigonometric Integrals - 7.2 Exercises - Page 525: 38

Answer

$$\frac{12}{35}$$

Work Step by Step

Given $$\int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta \cot ^{4} \theta d \theta$$ Since \begin{align*} \int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta \cot ^{4} \theta d \theta &=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta\csc ^{2} \theta \cot ^{4} \theta d \theta\\ &=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta(1+ \cot ^{2} \theta) \cot ^{4} \theta d \theta\\ &=\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta(\cot ^{4} \theta+ \cot ^{6} \theta) d \theta\\ &=\frac{1}{5}\cot ^{5} \theta+\frac{1}{7} \cot ^{7} \theta\bigg|_{\pi/4}^{\pi/2}\\ &=\frac{12}{35} \end{align*}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.