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: 12


$$5\ln |x-3|-3\ln |x-2|+C$$

Work Step by Step

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