## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{\left(x^{2}+1\right) d x}{\left(x^{3}+3 x+1\right)^{5}}
$$
## Solution

$$
\begin{aligned}
& \int \frac{\left(x^{2}+1\right) d x}{\left(x^{3}+3 x+1\right)^{5}}=\int \frac{\frac{1}{3} \cdot\left(3 x^{2}+3\right) d x}{\left(x^{3}+3 x+1\right)^{5}}=\int \frac{\frac{1}{3} \cdot d\left(x^{3}+3 x+1\right)}{\left(x^{3}+3 x+1\right)^{5}}= \\
& =\frac{1}{3} \cdot\left(-\frac{1}{4}\right) \cdot \frac{1}{\left(x^{3}+3 x+1\right)^{4}}+C=-\frac{1}{12} \cdot \frac{1}{\left(x^{3}+3 x+1\right)^{4}}+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-17$ »

Categories: Kuznetsov Problem Book Integrals Problem 3 | Integrals

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## Problem Kuznetsov Integrals 3-18

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