## Problem Statement

Calculate the indefinite integral:

$$
\int \operatorname{arctg} \sqrt{5 x-1} \, d x
$$
## Solution

$$
\int \operatorname{arctg} \sqrt{5 x-1} d x=
$$

Let:

$$
\begin{aligned}
& u=\operatorname{arctg} \sqrt{5 x-1} ; d u=\frac{1}{1+(\sqrt{5 x-1})^{2}} \cdot \frac{1}{2 \sqrt{5 x-1}} \cdot 5 d x= \\
& =\frac{1}{5 x} \cdot \frac{5}{2 \sqrt{5 x-1}} \cdot d x=\frac{d x}{2 x \sqrt{5 x-1}} \\
& d v=d x ; v=x
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =\operatorname{arctg} \sqrt{5 x-1} \cdot x-\int_{1} \frac{x \cdot d x}{2 x \sqrt{5 x-1}}=x \cdot \operatorname{arctg} \sqrt{5 x-1}-\int \frac{d x}{2 \sqrt{5 x-1}}= \\
& =x \cdot \operatorname{arctg} \sqrt{5 x-1}-\frac{1}{5} \cdot \sqrt{5 x-1}+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-16 »$ Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

- Last edited: 10:07, 24 March 2009.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 1-17

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