Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.5 The Method of Partial Fractions - Exercises - Page 423: 27


$$\frac{1}{2} \ln |2 x+5|+\frac{5}{(2 x+5)}-\frac{5}{4(2 x+5)^{2}}+C$$

Work Step by Step

Given $$\int \frac{4 x^{2}-20}{(2 x+5)^{3}} d x$$ Since \begin{aligned} \frac{4 x^{2}-20}{(2 x+5)^{3}} &=\frac{\mathrm{A}}{2x+5}+\frac{\mathrm{B}}{(2x+5)^2}+\frac{\mathrm{C}}{(2x+5)^3} \\ &=\frac{A(2 x+5)^{2}+B(2 x+5)+C}{(2 x+5)^{3}} \\ 4 x^{2}-20&= A(2 x+5)^{2}+B(2 x+5)+C \end{aligned} By comparing coefficients, we get \begin{align*} A&=1\\ B&=-10\\ C&=5 \end{align*} Hence \begin{aligned} \int \frac{4 x^{2}-20}{(2 x+5)^{3}} d x&=\int \frac{1}{2x+5}dx-10\int \frac{1}{(2x+5)^2}+5\int \frac{1}{(2x+5)^3}dx\\ &= \frac{1}{2} \ln |2 x+5|+\frac{5}{(2 x+5)}-\frac{5}{4(2 x+5)^{2}}+C \end{aligned}
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