## Problem Statement

Calculate the indefinite integral:

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

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

Under the integral, we have an improper fraction. Let's separate the integer part:

$$
\begin{array}{ll}
\frac{3 x^{3}+1}{} \\
\frac{3 x^{3}-3 x}{3 x+1} & \frac{x^{2}-1}{3 x}
\end{array}
$$

We get:

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

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

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

Then we get:

$$
=\int 3 x d x+\int\left(\frac{2}{x-1}+\frac{1}{x+1}\right) d x=\frac{3 x^{2}}{2}+2 \ln |x-1|+\ln |x+1|+C
$$

%tegraly_5-2»

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

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## Problem Kuznetsov Integrals 5-3

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