Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.1 - Using Basic Integration Formulas - Exercises 8.1 - Page 449: 37

Answer

$$\frac{{{\theta ^3}}}{3} - \frac{{{\theta ^2}}}{2} + \theta + \frac{5}{2}\ln \left| {2\theta - 5} \right| + C $$

Work Step by Step

$$\eqalign{ & \int {\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}}} d\theta \cr & {\text{Use long division on }}\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}} \cr & \frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}} = {\theta ^2} - \theta + 1 + \frac{5}{{2\theta - 5}} \cr & {\text{Then}} \cr & \int {\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}}} d\theta = \int {\left( {{\theta ^2} - \theta + 1 + \frac{5}{{2\theta - 5}}} \right)} d\theta \cr & = \int {{\theta ^2}} d\theta - \int \theta d\theta + \int {d\theta } + \int {\frac{5}{{2\theta - 5}}} d\theta \cr & {\text{Integrating}} \cr & = \frac{{{\theta ^3}}}{3} - \frac{{{\theta ^2}}}{2} + \theta + \frac{5}{2}\ln \left| {2\theta - 5} \right| + C \cr} $$
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