Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises: 87

Answer

$\frac{2}{3}(1+e^x)^\frac{3}{2}+C$

Work Step by Step

Let $u=1+e^x$. Then $du=e^x dx$. $\int e^x\sqrt{1+e^x}dx$ $=\int \sqrt{1+e^x}e^xdx$ $=\int \sqrt{u}du$ $=\int u^\frac{1}{2}du$ $=\frac{u^\frac{3}{2}}{\frac{3}{2}}+C$ $=\frac{2}{3}u^\frac{3}{2}+C$ $=\frac{2}{3}(1+e^x)^\frac{3}{2}+C$
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