Calculus: Early Transcendentals 8th Edition

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

Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises: 29

Answer

$$\int 5^t\sin(5^t)dt=-\frac{\cos(5^t)}{\ln5}+C$$

Work Step by Step

$$A=\int 5^t\sin(5^t)dt$$ Let $u=5^t$. We would have $du=(5^t)'dt=5^t\ln 5dt$. Therefore, $5^tdt=\frac{1}{\ln 5}du$ Substitute into $A$, we have $$A=\frac{1}{\ln5}\int\sin udu$$ $$A=\frac{1}{\ln5}(-\cos u)+C$$ $$A=-\frac{\cos(5^t)}{\ln5}+C$$
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