Calculus: Early Transcendentals (2nd Edition)

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

Chapter 6 - Applications of Integration - Review Exercises: 62

Answer

$${\sinh ^{ - 1}}\frac{{{e^x}}}{2} + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{{{e^x}}}{{\sqrt {{e^{2x}} + 4} }}} dx \cr & {\text{rewriting radicand}} \cr & \int {\frac{{{e^x}}}{{\sqrt {{{\left( {{e^x}} \right)}^2} + {{\left( 2 \right)}^2}} }}} dx \cr & u = {e^x},{\text{ }}du = {e^x}dx \cr & {\text{find the antiderivative using the theorem 6}}{\text{.12}}{\text{, formula 2}} \cr & \int {\frac{{du}}{{\sqrt {{u^2} + {a^2}} }}} = {\sinh ^{ - 1}}\frac{u}{a} + C \cr & so \cr & = {\sinh ^{ - 1}}\frac{{{e^x}}}{2} + C \cr} $$
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