## Problem Statement

Calculate the indefinite integral:

$$
\int \ln \left(x^{2}+4\right) d x
$$
## Solution

$$
\int \ln \left(x^{2}+4\right) d x=
$$

Let:

$$
\begin{aligned}
& u=\ln \left(x^{2}+4\right) ; d u=\frac{1}{x^{2}+4} \cdot 2 x \cdot d x=\frac{2 x d x}{x^{2}+4} \\
& d v=d x ; v=x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\ln \left(x^{2}+4\right) \cdot x-\int \frac{2 x^{2}}{x^{2}+4} d x=x \cdot \ln \left(x^{2}+4\right)-2 \int \frac{x^{2}+4-4}{x^{2}+4} d x= \\
& =x \cdot \ln \left(x^{2}+4\right)-2 \int d x+8 \int \frac{1}{x^{2}+4} d x=x \cdot \ln \left(x^{2}+4\right)-2 x+8 \cdot \frac{1}{2} \cdot \operatorname{arctg} \frac{x}{2}+C= \\
& =x \cdot \ln \left(x^{2}+4\right)-2 x+4 \operatorname{arctg} \frac{x}{2}+C
\end{aligned}
$$

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

Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

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## Problem Kuznetsov Integrals 1-9

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