Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises: 32

Answer

\[ = - \frac{{{x^3}}}{3} + 4x - 5{\tan ^{ - 1}}\frac{x}{2} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{6 - {x^4}}}{{{x^2} + 4}}dx} \hfill \\ \hfill \\ {\text{using}}\,\,{\text{the}}\,\,{\text{long}}\,\,{\text{division}} \hfill \\ \hfill \\ = \int_{}^{} {\,\left( { - {x^2} + 4 - \frac{{10}}{{{x^2} + 4}}} \right)dx} \hfill \\ \hfill \\ Usi\,ng\,\int_{}^{} {\frac{{dx}}{{{a^2} + {x^2}}} = \frac{1}{a}\,{{\tan }^{ - 1}}\frac{x}{a}} \,\,for\,the\,last\,term \hfill \\ \hfill \\ = - \frac{{{x^3}}}{3} + 4x - 10\frac{1}{2}{\tan ^{ - 1}}\frac{x}{2} + C \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ = - \frac{{{x^3}}}{3} + 4x - 5{\tan ^{ - 1}}\frac{x}{2} + C \hfill \\ \end{gathered} \]
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