## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{4 \operatorname{arctg} x - x}{1 + x^{2}} \, dx
$$
## Solution

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

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�

$\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \_\% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD}$

$\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-18 »$

Categories: Kuznetsov's Problem Book Integrals Problem $3 \mid$ Integrals

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## Problem Kuznetsov Integrals 3-19

## Material from PlusPi

## Contents

- 1 Problem Statement
- 2 Solution
- 2.1 Solution #1
- 2.2 Solution #2