Calculus (3rd Edition)

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

Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 275: 42


$$\ln |x+5|-\frac{25}{2(x+5)^{2}}+\frac{10}{x+5}+c$$

Work Step by Step

Given $$ \int \frac{x^2d x}{(x+5)^{3}}$$ Let $$u=x+5 \ \ \ \Rightarrow \ \ du= dx $$ then \begin{align*} \int \frac{x^2d x}{(x+5)^{3}} &=\int \frac{(u-5)^2d u}{u^{3}} \\ &=\int \frac{(u^2-10u+25)d u}{u^{3}} \\ &= \int \frac{du}{u} -10\int \frac{du}{u^2}+25\int \frac{du}{u^3}\\ &=\ln |u| +\frac{10}{u}-\frac{25}{2u^2}+c\\ &=\ln |x+5|-\frac{25}{2(x+5)^{2}}+\frac{10}{x+5}+c \end{align*}
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