Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 83


$2\sqrt{e^x+1}+c $

Work Step by Step

Recall that $(e^x)'=e^x$ Let $ u=e^x $, then $ du=e^x dx $ and hence we have $$ \int \frac{e^x}{ \sqrt {e^x+1}} dx=\int \frac{1}{ \sqrt {u+1}} du\\ =\int (u+1)^{-1/2}du=\frac{1}{1/2} (u+1)^{1/2}+c=2\sqrt{e^x+1}+c . $$
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