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: 76

Answer

$$\int\frac{\sin(\ln x)}{x}dx=-\cos(\ln x)+c$$

Work Step by Step

To evaluate the integral $$\int\frac{\sin(\ln x)}{x}dx$$ we will use substitution $\ln x=t$ which gives us $\frac{dx}{x}=dt$. Putting this into the integral we get: $$\int\frac{\sin(\ln x)}{x}dx=\int\sin tdx=-\cos t+c$$ where $c$ is arbitrary constant. Now we have to express solution in terms of $x$: $$\int\frac{\sin(\ln x)}{x}dx=-\cos t+c=-\cos(\ln x)+c$$
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