Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.5 The Method of Partial Fractions - Exercises - Page 424: 47


$$\frac{1}{2} \tan ^{-1}\left(x^{2}\right)+C$$

Work Step by Step

Given $$ \int \frac{x}{x^{4}+1} d x$$ Let $$u=x^2 \ \ \ \ \ \to \ \ \ \ du=2dx $$ Then \begin{align*} \int \frac{x}{x^{4}+1} d x&=\frac{1}{2} \int \frac{1}{u^{2}+1} d u\\ &=\frac{1}{2} \tan ^{-1} u+C\\ &=\frac{1}{2} \tan ^{-1}\left(x^{2}\right)+C \end{align*}
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