Calculus: Early Transcendentals (2nd Edition)

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

Chapter 6 - Applications of Integration - 6.8 Logarithmic and Exponential - 6.8 Exercises: 17

Answer

$$\frac{1}{2}\ln \left( {4 + {e^{2x}}} \right) + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{{{e^{2x}}}}{{4 + {e^{2x}}}}dx} \cr & {\text{substitute }}u = 4 + {e^{2x}},{\text{ }}du = 2{e^{2x}}dx \cr & \int {\frac{{{e^{2x}}}}{{4 + {e^{2x}}}}dx} = \int {\frac{{\left( {1/2} \right)}}{u}du} \cr & = \frac{1}{2}\int {\frac{1}{u}} du \cr & {\text{find the antiderivative}} \cr & = \frac{1}{2}\ln \left| u \right| + C \cr & {\text{with }}u = 4 + {e^{2x}} \cr & = \frac{1}{2}\ln \left| {4 + {e^{2x}}} \right| + C \cr & = \frac{1}{2}\ln \left( {4 + {e^{2x}}} \right) + C \cr} $$
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