Calculus: Early Transcendentals (2nd Edition)

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

Chapter 5 - Integration - Review Exercises: 30

Answer

\[ = {\tan ^{ - 1}}\,\left( 2 \right) - \frac{\pi }{4}\]

Work Step by Step

\[\begin{gathered} \int_0^{\ln 2} {\frac{{{e^x}}}{{1 + {e^{2x}}}}} \hfill \\ \hfill \\ so\,,\,\int_0^{\ln 2} {\frac{{{e^x}}}{{1 + \,{{\left( {{e^x}} \right)}^2}}}} \,\,dx \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \,\,\left[ {{{\tan }^{ - 1}}\,{e^x}} \right]_0^{\ln 2} \hfill \\ \hfill \\ Fundamental\,\,theorem \hfill \\ \hfill \\ \,\,\left[ {{{\tan }^{ - 1}}{e^{\ln 2}}} \right] - \,\,\left[ {{{\tan }^{. - }}{e^0}} \right] \hfill \\ \hfill \\ {\text{Simplify}} \hfill \\ \hfill \\ = {\tan ^{ - 1}}\,\left( 2 \right) - {\tan ^{ - 1}}\,\left( 1 \right) \hfill \\ \hfill \\ = {\tan ^{ - 1}}\,\left( 2 \right) - \frac{\pi }{4} \hfill \\ \end{gathered} \]
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