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 - Page 347: 79


$$\int\cot xdx=\ln|\sin x|+c$$

Work Step by Step

To evaluate the integral $$\int\cot xdx=\int\frac{\cos x}{\sin x}dx$$ we will use substitution $\sin x=t$ which gives us $\cos xdx=dt$. Putting this into the integral we get: $$\int\frac{\cos x}{\sin x}dx=\int\frac{1}{t}dt=\ln|t|+c$$ where $c$ is arbitrary constant. Now we have to express solution in terms of $x$: $$\int\cot xdx=\ln|t|+c=\ln|\sin x|+c$$
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