## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{x}{\sqrt{x^{4}-x^{2}-1}} d x
$$
## Solution

$$
\begin{aligned}
& \int \frac{x}{\sqrt{x^{4}-x^{2}-1}} d x=\int \frac{\frac{1}{2} \cdot d\left(x^{2}-\frac{1}{2}\right)}{\sqrt{\left(x^{2}-\frac{1}{2}\right)^{2}-\frac{5}{4}}}=\frac{1}{2} \cdot \ln \left|x^{2}-\frac{1}{2}+\sqrt{\left(x^{2}-\frac{1}{2}\right)^{2}-\frac{5}{4}}\right|+C= \\
& =\frac{1}{2} \cdot \ln \left|x^{2}-\frac{1}{2}+\sqrt{x^{4}-x^{2}-1}\right|+C
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/%D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2_%D0%98%D0%BD%D1%82 $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} 3-14$ " Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

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## Problem Kuznetsov Integrals 3-15

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