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

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

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*}