Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Practice Exercises - Page 517: 21


$\dfrac{x^2}{2}+\dfrac{4}{3} \ln |x+2|+\dfrac{2}{3} \ln |x-1|+C$

Work Step by Step

Apply the integration by parts formula as follows: $\int a'(x) b(x)=a(x) b(x)-\int a(x) b'(x)dx$ Re-write the integral into partial fractions. $\int \dfrac{x^3+x^2}{x^2+x-2}dx=\dfrac{(x^3+x^2-2x+2x) dx}{x^2+x-2}$ Now, $\int \dfrac{x^3+x^2-2x}{x^2+x-2} dx+ \int \dfrac{2x}{x^2+x-2}dx=\int x \space dx+\int \dfrac{\frac{4}{3}}{x+2}+\dfrac{\dfrac{2}{3}}{x-1} dx \\=\dfrac{x^2}{2}+\dfrac{4}{3} \ln |x+2|+\dfrac{2}{3} \ln |x-1|+C$
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