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: 47


$$\frac{1}{2}\left (\sin t+\frac{1}{2}\sin 2(\sin t) \right)+c$$

Work Step by Step

Given $$\int \cos^2(\sin t)\cos tdt $$ Let $$ u =\sin t\ \ \ \to \ \ \ du=\cos tdt $$ Then \begin{align*} \int \cos^2(\sin t)\cos tdt &=\int \cos^2udu\\ &=\frac{1}{2}\int (1+\cos 2u )du\\ &=\frac{1}{2}\left (u+\frac{1}{2}\sin 2u \right)+c\\ &=\frac{1}{2}\left (\sin t+\frac{1}{2}\sin 2(\sin t) \right)+c \end{align*}
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