Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 7 - Principles Of Integral Evaluation - 7.1 An Overview Of Integration Methods - Exercises Set 7.1 - Page 491: 27

Answer

$$ - \frac{1}{2}\cos \left( {{x^2}} \right) + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{x}{{\csc \left( {{x^2}} \right)}}} dx \cr & {\text{trigonometric identity sin}}\phi = \frac{1}{{\csc \phi }} \cr & = \int {\sin \left( {{x^2}} \right)x} dx \cr & {\text{substitute }}u = {x^2}{\text{ }} \cr & du = 2xdx \cr & \frac{1}{2}du = xdx \cr & = \frac{1}{2}\int {\sin u} du \cr & {\text{find the antiderivative}} \cr & = \frac{1}{2}\left( { - \cos u} \right) + C \cr & = - \frac{1}{2}\cos u + C \cr & {\text{write in terms of }}x,{\text{ replace }}u = {x^2}{\text{ }} \cr & = - \frac{1}{2}\cos \left( {{x^2}} \right) + C \cr} $$
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