Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.6 Other Integration Strategies - 7.6 Exercises: 31

Answer

\[\ln \left| {{e^x} + \sqrt {{e^{2x}} + 4} } \right| + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{{e^x}}}{{\sqrt {{e^{2x}} + 4} }}} \hfill \\ \hfill \\ rewrite\,\,the\,\,{\text{denominator}} \hfill \\ \hfill \\ = \int_{}^{} {\frac{{{e^x}dx}}{{\sqrt {\,{{\left( {{e^x}} \right)}^2} + {2^2}} }}} \hfill \\ \hfill \\ Use\,\,the\,\,formula\,\,\,\int_{}^{} {\frac{{du}}{{\sqrt {{u^2} + {a^2}} }} = } \ln \left| {u + \sqrt {{u^2} + {a^2}} } \right| + C \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \int_{}^{} {\frac{{{e^x}}}{{\sqrt {{e^{2x}} + 4} }}} = \ln \left| {{e^x} + \sqrt {{e^{2x}} + 4} } \right| + C \hfill \\ \end{gathered} \]
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