Calculus with Applications (10th Edition)

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

Chapter 7 - Integration - 7.2 Substitution - 7.2 Exercises: 17

Answer

\[\frac{1}{2}\ln \,\left( {{t^2} + 2} \right) + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{t}{{{t^2} + 2}}dt} \hfill \\ Let\,\,u = {t^2} + 2\,\,\,so\,\,that \hfill \\ \,\,\,\,du = 2tdt \hfill \\ \int_{}^{} {\frac{t}{{{t^2} + 2}}dt\,\, = \frac{1}{2}\int_{}^{} {\frac{{2t}}{{{t^2} + 2}}dt} } \hfill \\ \frac{1}{2}\int_{}^{} {\frac{{du}}{u}} \hfill \\ Integrating \hfill \\ \frac{1}{2}\ln \left| u \right| + C \hfill \\ Substituting\,\,u = {t^2} + 2\,\,for\,\,u\,\,gives \hfill \\ \frac{1}{2}\ln \,\left( {{t^2} + 2} \right) + C \hfill \\ \hfill \\ \end{gathered} \]
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