Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises: 72

Answer

$$\int e^{\cos t}\sin tdt=-e^{\cos t}+c$$

Work Step by Step

To evaluate the integral $$\int e^{\cos t}\sin tdt$$ we will use substitution $\cos t=z$ which gives us $-\sin tdt=dz\Rightarrow \sin tdt=-dz$. Putting this into the integral we get: $$\int e^{\cos t}\sin tdt=\int e^z(-dz)=-e^z+c$$ where $c$ is arbitrary constant. Now we have to express solution in terms of $t$: $$\int e^{\cos t}\sin tdt=-e^z+c=-e^{\cos t}+c$$
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