University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Practice Exercises - Page 474: 37


$\dfrac{\cos^7 x}{7}-\dfrac{\cos^5 x}{5}+c$

Work Step by Step

Consider the integral $\int \sin^3 x \cos^4 x dx$ The given integral can be re-written as: $\int \sin^3 x \cos^4 x dx=\int \sin^2 x \cos^4 x (\sin x dx)=\int (1-cos^2 x) \cos^4 x (\sin x dx)$ Plug in $a= \cos x \implies da= -\sin x dx$ This implies that $\int (a^2-1) a^4 da =\int a^6-a^4 da$ We use the formula: $\int x^n dx=\int\dfrac{x^{n+1}}{n+1} dx$ $\dfrac{a^7}{7}-\dfrac{a^5}{5}+c=\dfrac{\cos^7 x}{7}-\dfrac{\cos^5 x}{5}+c$
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