## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{x^{3}}{\left(x^{2}+1\right)^{2}} d x
$$
## Solution

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-9$ "

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

- Last edited: March 1, 2009, at 17:04.
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## Problem Kuznetsov Integrals 3-10

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