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: 47


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

Work Step by Step

Given $$ \int \frac{\left(x^{2}-x\right) d x}{(x+2)^{3}} $$ Since \begin{align*} \frac{\left(x^{2}-x\right) }{(x+2)^{3}}&= \frac{A}{(x+2)^3}+ \frac{B}{(x+2)^2}+ \frac{C}{(x+2) }\\ &=\frac{ A+B(x+2)+C(x+2)^2 }{(x+2)^{3}}\\ x^2-x&= A+B(x+2)+C(x+2)^2 \end{align*} Then by comparing the coefficients, we get $$ C= 1,\ \ B=-5,\ \ A=6$$ Hence \begin{align*} \int \frac{\left(x^{2}-x\right) }{(x+2)^{3}}dx&=\int \frac{6}{(x+2)^3}dx+ \int \frac{-5}{(x+2)^2}dx+ \int\frac{1}{(x+2) }dx \\ &=-\frac{3}{\left(x+2\right)^2} +\frac{5}{x+2} +\ln \left|x+2\right|+C \end{align*}
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