Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises: 25

Answer

$$\int e^x\sqrt{1+e^x}dx=\frac{2}{3}\sqrt{(1+e^x)^3}+C$$

Work Step by Step

$$A=\int e^x\sqrt{1+e^x}dx$$ Let $u=1+e^x$. We would have $du=e^xdx$. Also, $\sqrt{1+e^x}=\sqrt u=u^{1/2}$ Substitute into $A$, we have $$A=\int u^{1/2}du$$ $$A=\frac{u^{3/2}}{\frac{3}{2}}+C$$ $$A=\frac{2\sqrt{u^3}}{3}+C$$ $$A=\frac{2}{3}\sqrt{(1+e^x)^3}+C$$
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