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: 2


$$\int x(2x^2+3)^4dx=\frac{(2x^2+3)^5}{20}$$

Work Step by Step

To evaluate the integral $\int x(2x^2+3)^4dx$ we will use substitution $u=2x^2+3$ which gives us $du=4xdx\Rightarrow \frac{1}{4}du=xdx.$ Putting this into our integral we get: $$\int(2x^2+3)^4xdx=\int u^4\cdot\frac{1}{4}du=\frac{1}{4}\int u^4du=\frac{1}{4}\frac{u^5}{5}=\frac{u^5}{20}$$ Now we have to express our solution in terms of $x$ by expressing $u$ in terms of $x$: $$\int x(2x^2+3)^4dx=\frac{u^5}{20}=\frac{(2x^2+3)^5}{20}$$
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