Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 27


$$\frac{1}{\pi} \left(\frac{1}{5}\sin^5(\pi\theta)-\frac{1}{7}\sin^7 (\pi\theta) \right)+c $$

Work Step by Step

Given $$ \int \cos ^{3}(\pi \theta) \sin ^{4}(\pi \theta) d \theta $$ Let $$u=\pi \theta\ \ \Rightarrow \ \ du =\pi d\theta$$ Then \begin{align*} \int \cos ^{3}(\pi \theta) \sin ^{4}(\pi \theta) d \theta&=\frac{1}{\pi} \int \cos ^{3} u \sin ^{4} u d u\\ &=\frac{1}{\pi} \int\left(1-\sin ^{2} u\right) \sin ^{4} u \cos u d u\\ &=\frac{1}{\pi} \int\left(\sin^4u-\sin ^{6} u\right) \cos u d u\\ &=\frac{1}{\pi} \left(\frac{1}{5}\sin^5u-\frac{1}{7}\sin^7 u \right)+c \\ &=\frac{1}{\pi} \left(\frac{1}{5}\sin^5(\pi\theta)-\frac{1}{7}\sin^7 (\pi\theta) \right)+c \end{align*}
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