Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.9 Antiderivatives - 4.9 Exercises: 44

Answer

$$ - \frac{1}{3}\csc 3\varphi + C$$

Work Step by Step

$$\eqalign{ & \int {\csc 3\varphi } \cot 3\varphi d\varphi \cr & {\text{use the formula for indefinite integrals of trigonometric functions}} \cr & \int {\csc ax} \cot axdx = - \frac{1}{a}\csc ax + C \cr & {\text{letting }}a = 3,{\text{ and }}x = \varphi ,{\text{ we have}} \cr & = - \frac{1}{3}\csc 3\varphi + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{d\varphi }}\left( { - \frac{1}{3}\csc 3\varphi + C} \right) \cr & {\text{ = }}\frac{d}{{d\varphi }}\left( { - \frac{1}{3}\csc 3\varphi } \right) + \frac{d}{{d\varphi }}\left( C \right) \cr & = - \frac{1}{3}\left( { - \csc 3\varphi \cot 3\varphi } \right)\left( 3 \right) + 0 \cr & = \csc 3\varphi \cot 3\varphi \cr} $$
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