Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 7 - Integration - 7.1 Antiderivatives - 7.1 Exercises: 33

Answer

\[\frac{1}{4}\ln \left| t \right| + \frac{{{t^3}}}{6} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{1 + 2{t^3}}}{{4t}}dt} \hfill \\ rewrite\,\,the\,\,integrand\,\,as\,\,follows \hfill \\ \int_{}^{} {\,\left( {\frac{1}{{4t}} + \frac{{2{t^3}}}{{4t}}} \right)dt} \hfill \\ \int_{}^{} {\frac{1}{{4t}}dt + \int_{}^{} {\frac{{{t^2}}}{2}dt} } \hfill \\ Use\,\,indefinite\,\,integrals \hfill \\ \int_{}^{} {{t^{ - 1}}dt = \ln \left| t \right| + C\,\,,\,\,\int_{}^{} {{t^n}dt} = \frac{{{t^{n + 1}}}}{{n + 1}} + C} \hfill \\ \frac{1}{4}\ln \left| t \right| + \frac{1}{2}\,\left( {\frac{{{t^3}}}{3}} \right) + C \hfill \\ Simplifying \hfill \\ \frac{1}{4}\ln \left| t \right| + \frac{{{t^3}}}{6} + C \hfill \\ \end{gathered} \]
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