## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{x^{3}}{x^{2}+1} d x
$$
## Solution

$$
\begin{aligned}
& \int \frac{x^{3}}{x^{2}+1} d x=\int \frac{x\left(x^{2}+1\right)-x}{x^{2}+1} d x=\int x d x-\int \frac{x}{x^{2}+1}= \\
& =\frac{1}{2} x^{2}-\frac{1}{2} \int \frac{d\left(x^{2}+1\right)}{x^{2}+1}=\frac{1}{2}\left(x^{2}-\ln \left(x^{2}+1\right)\right)+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-29$ "

Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

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## Problem Kuznetsov Integrals 3-30

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