## Problem Statement

Calculate the indefinite integral:

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

Let's introduce the substitution

$$
\begin{aligned}
& x=\frac{1}{t} \\
& d x=-\frac{1}{t^{2}} d t
\end{aligned}
$$

Then

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-1$ " Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

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## Problem Kuznetsov Integrals 3-2

## Material from PlusPi

## Contents

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