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 460: 41


$$\ln |x+2|+\frac{5}{x+2}-\frac{3}{(x+2)^{2}}+C$$

Work Step by Step

Given $$\int \frac{\left(x^{2}-x\right) d x}{(x+2)^{3}}$$ Since \begin{aligned} \frac{\left(x^{2}-x\right)}{(x+2)^{3}} &=\frac{A}{(x+2)}+\frac{B}{(x+2)^{2}}+\frac{C}{(x+2)^{3}} \\ &=\frac{A(x+2)^{2}+B(x+2)+C}{(x+2)^{3}} \\ x^{2}-x &=A(x+2)^{2}+B(x+2)+C \end{aligned} Then \begin{align*} \text{at } x&=-2 \ \ \ \ \to C=6 \\ \text{coefficient of } x^2&\to \ \ \ \ \to A=1 \\ \text{coefficient of } x&\to \ \ \ \ \to B=-5 \end{align*} Then \begin{aligned} \int \frac{\left(x^{2}-x\right) d x}{(x+2)^{3}} &=\int \frac{1}{(x+2)} d x-5 \int \frac{1}{(x+2)^{2}} d x+6 \int \frac{1}{(x+2)^{3}} d x \\ &=\int \frac{1}{(x+2)} d x-5 \int(x+2)^{-2} d x+6 \int(x+2)^{-3} d x \\ &=\ln |x+2|+\frac{5}{x+2}-\frac{3}{(x+2)^{2}}+C \end{aligned}
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