## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{4 x^{4}+2 x^{2}-x-3}{x(x-1)(x+1)} d x
$$
## Solution

$$
\int \frac{4 x^{4}+2 x^{2}-x-3}{x(x-1)(x+1)} d x=\int \frac{4 x^{4}+2 x^{2}-x-3}{x^{3}-x} d x=
$$

The integrand is an improper fraction. Let's separate the integer part:

$$
\begin{array}{ll}
4 x^{4}+2 x^{2}-x-3 & \left\lvert\, \frac{x^{3}-x}{4 x}\right. \\
\frac{4 x^{4}-4 x^{2}}{6 x^{2}-x-3}
\end{array}
$$

We get:

$$
=\int\left(4 x+\frac{6 x^{2}-x-3}{x^{3}-x}\right) d x=\int 4 x d x+\int \frac{6 x^{2}-x-3}{x(x-1)(x+1)} d x=
$$

Let's decompose the proper rational fraction into partial fractions using the method of undetermined coefficients:

$$
\begin{aligned}
& \frac{6 x^{2}-x-3}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}= \\
& =\frac{A(x-1)(x+1)+B x(x+1)+C x(x-1)}{x(x-1)(x+1)}= \\
& =\frac{A\left(x^{2}-1\right)+B\left(x^{2}+x\right)+C\left(x^{2}-x\right)}{x(x-1)(x+1)}= \\
& =\frac{(A+B+C) x^{2}+(B-C) x-A}{x(x-1)(x+1)}= \\
& \left\{\begin{array}{l}
A+B+C=6 \\
B-C=-1 \\
-A=-3
\end{array}\right.
\end{aligned}
$$

$$
\left\{\begin{array}{l}
B+C=3 \\
B-C=-1 \\
A=3
\end{array}\right.
$$

Add the first row to the second:

$$
\begin{aligned}
& \left\{\begin{array}{l}
B+C=3 \\
2 B=2 \\
A=3
\end{array}\right. \\
& \left\{\begin{array}{l}
C=2 \\
B=1 \\
A=3
\end{array}\right. \\
& \frac{6 x^{2}-x-3}{x(x-1)(x+1)}=\frac{3}{x}+\frac{1}{x-1}+\frac{2}{x+1}
\end{aligned}
$$

Then we get:

$$
\begin{aligned}
& =\int 4 x d x+\int\left(\frac{3}{x}+\frac{1}{x-1}+\frac{2}{x+1}\right) d x= \\
& =2 x^{2}+3 \cdot \ln |x|+\ln |x-1|+2 \cdot \ln |x+1|+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+5-24$ »

Categories: Kuznetsov's Problem Book Integrals Problem 5 | Integrals

- Last modified: 20:51, 6 April 2009.
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## Problem Kuznetsov Integrals $5-25$

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