Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 459: 2


$$x-2 \ln |x+2|+C $$

Work Step by Step

Given $$\int \frac{x}{x+2} d x $$ Let $ u= x+2\ \ \to du =dx$ Then\begin{aligned} \int \frac{x}{x+2} d x &=\int \frac{u-2}{u} d u \\ &=\int \frac{u}{u} d u-\int \frac{2}{u} d u \\ &=\int 1 d u-\int \frac{2}{u} d u \\ &=u-2 \ln |u|+C \\ &=x+2-2 \ln |x+2|+C_{1} \\ &=x-2 \ln |x+2|+C \end{aligned} Or by partial fractions \begin{aligned} \int \frac{x}{x+2} d x &=\int 1 d x-\int \frac{2}{x+2} d x \\ &=x-2 \ln |x+2|+C \end{aligned}
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