Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 36


$$\frac{1}{3}\left( (x^3-1)+\ln{|x^3-1|} \right)+C$$

Work Step by Step

Given $$\int \frac{x^{5}}{x^{3}-1} d x$$ Let $$u=x^3-1\ \ \ \ \ \ \ \ du=3x^2dx $$Then \begin{align*} \int \frac{x^{5}}{x^{3}-1} d x&=\frac{1}{3}\int \frac{u+1}{u} du\\ &=\frac{1}{3}\int\left( 1+\frac{1}{u} \right)du\\ &=\frac{1}{3}\left( u+\ln{u} \right)+C\\ &=\frac{1}{3}\left( (x^3-1)+\ln{|x^3-1|} \right)+C \end{align*}
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