Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 29


$$\int x(2x+5)^8dx=\frac{(2x+5)^9(18x-5)}{360}+c$$

Work Step by Step

To solve the integral $\int x(2x+5)^8dx$ we will use substitution $t=2x+5$ which gives us $x=\frac{x-5}{2}$ and $2dx=dt\Rightarrow dx=\frac{1}{2}dt.$ Putting this into the integral we get: $$\int x(2x+5)^8dx=\int\frac{t-5}{2}t^8\frac{dt}{2}=\frac{1}{4}\int t^9dt-\frac{5}{4}\int t^8dt= \frac{1}{4}\frac{t^{10}}{10}-\frac{5}{4}\frac{t^9}{9}+c=\frac{t^9}{4}(\frac{t}{10}-\frac{5}{9})+c$$ where $c$ is arbitrary constant. Now we have to express solution in terms of $x$ by expressing $t$ in terms of $x$: $$\int x(2x+5)^8dx=\frac{t^9}{4}(\frac{t}{10}-\frac{5}{9})+c=\frac{(2x+5)^9}{4}(\frac{2x+5}{10}-\frac{5}{9})+c=\frac{(2x+5)^9}{4}\cdot\frac{18x+45-50}{90}+c=\frac{(2x+5)^9(18x-5)}{360}+c$$
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