Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 57

Answer

\[ = \ln \left| {\sec \,\left( {{e^x} + 1} \right) + \tan\,\,\left( {{e^x} + 1} \right)} \right| + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{e^x}\sec \,\left( {{e^x} + 1} \right)\,dx} \hfill \\ \hfill \\ set\,\,\,{e^x} + 1 = u\,\,\,\,\,\,\,then\,\,\,\,\,\,{e^x}dx = du \hfill \\ \hfill \\ = \int_{}^{} {\sec u\,\,dx} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \ln \left| {\sec u + tan\,u} \right| + C \hfill \\ \hfill \\ \,substitute\,\,back\,\,u = {e^x} + 1 \hfill \\ \hfill \\ = \ln \left| {\sec \,\left( {{e^x} + 1} \right) + \tan\,\,\left( {{e^x} + 1} \right)} \right| + C \hfill \\ \end{gathered} \]
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