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: 44

Answer

$$\int\frac{x}{1+x^4}dx=\frac{\tan^{-1}(x^2)}{2}+C$$

Work Step by Step

$$A=\int\frac{x}{1+x^4}dx$$ In the substitution exercise with a fraction like this, the first thing you should think is to find out if you can find some substitution strategy to remove the numerator and change the fraction to familiar forms. Here we would do exactly like that: Let $u=x^2$ Then we have $du=2xdx$. That means $xdx=\frac{1}{2}du$ Also, $1+x^4=1+(x^2)^2=1+u^2$ Substitute into $A$: $$A=\frac{1}{2}\int\frac{1}{1+u^2}du$$ $$A=\frac{1}{2}\tan^{-1}u+C$$ $$A=\frac{\tan^{-1}(x^2)}{2}+C$$
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