Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises: 31

Answer

$$\int\frac{(\arctan x)^2}{x^2+1}dx=\frac{(\arctan x)^3}{3}+C$$

Work Step by Step

$$A=\int\frac{(\arctan x)^2}{x^2+1}dx$$ Let $u=\arctan x=\tan^{-1}x$. We would have $du=\frac{1}{x^2+1}dx$. Substitute into $A$, we have $$A=\int u^2du$$ $$A=\frac{u^3}{3}+C$$ $$A=\frac{(\arctan x)^3}{3}+C$$
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