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 404: 68


$$ \frac{1}{4}\left(\frac{3}{4}(2x)+\sin 2x +\frac{1}{4}\sin 2x \cos 2x\right)+c$$

Work Step by Step

\begin{aligned} \int \cos ^{4} x d x &=\int\left(\frac{1}{2}(1+\cos 2 x)\right)^{2} d x \\ &=\frac{1}{4} \int(1+\cos 2 x)^{2} d x \\ &=\frac{1}{4} \int\left(1+2 \cos 2 x+\cos ^{2} 2 x\right) d x \\ &=\frac{1}{4} \int\left(1+2 \cos 2 x+\frac{1}{2}+ \frac{1}{2}\cos 4 x\right) dx\\ &=\frac{1}{4}\left(\frac{3}{2}x+\sin 2x +\frac{1}{8}\sin 4x \right)+c\\ &= \frac{1}{4}\left(\frac{3}{4}(2x)+\sin 2x +\frac{1}{4}\sin 2x \cos 2x\right)+c \end{aligned}
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