Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.5 - Strategy for Integration - 7.5 Exercises - Page 507: 13


$-\frac{1}{5}\cos^5 t+\frac{2}{7}\cos^7 t-\frac{1}{9}\cos^9 t+C$

Work Step by Step

$\int\sin^5 t\cos^4 t\ dt$ Since the power of sine is odd, save a factor of $\sin t$ and express the remaining factors in terms of $\cos t$: $=\int\sin^4 t\cos^4 t\sin t\ dt$ $=\int(\sin^2 t)^2\cos^4 t\sin t\ dt$ $=\int(1-\cos^2 t)^2\cos^4 t\sin t\ dt$ Let $u=\cos t$. Then $du=-\sin t\ dt$, and $\sin t\ dt=-du$. $=\int(1-u^2)^2 u^4*(-1)\ du$ $=-\int(1-2u^2+u^4)u^4\ du$ $=-\int(u^4-2u^6+u^8)\ du$ $=-(\frac{1}{5}u^5-\frac{2}{7}u^7+\frac{1}{9}u^9+C)$ $=-(\frac{1}{5}\cos^5 t-\frac{2}{7}\cos^7 t+\frac{1}{9}\cos^9 t+C)$ $=\boxed{-\frac{1}{5}\cos^5 t+\frac{2}{7}\cos^7 t-\frac{1}{9}\cos^9 t+C}$
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