## Problem Statement

Calculate the indefinite integral:

$$
\int \ln \left(4 x^{2}+1\right) d x
$$
## Solution

$$
\int \ln \left(4 x^{2}+1\right) d x=
$$

Let:

$$
\begin{aligned}
& u=\ln \left(4 x^{2}+1\right) ; d u=\frac{1}{4 x^{2}+1} \cdot 8 x \cdot d x=\frac{8 x d x}{4 x^{2}+1} \\
& d v=d x ; v=x
\end{aligned}
$$

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

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

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%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_1-10" Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

- Last edited on this page: 18:55, 17 February 2009.
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## Problem Kuznetsov Integrals 1-10

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