Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 38


$$\frac{2}{3}(x+2)^{3/2} -4\sqrt{x+2}+C$$

Work Step by Step

Given $$\int \frac{x}{\sqrt{x+2}} d x$$ Let $$ u^2=x+2\ \ \ \ \ 2udu=dx$$ Then \begin{align*} \int \frac{x}{\sqrt{x+2}} d x&=2\int \frac{u(u^2-2)du}{u} \\ &= 2\int (u^2-2)du\\ &=\frac{2}{3}u^3 -4u+C\\ &=\frac{2}{3}(x+2)^{3/2} -4\sqrt{x+2}+C \end{align*}
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