## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{3 x^{3}+9 x^{2}+10 x+2}{(x-1)(x+1)^{3}} d x
$$
## Solution

$$
\int \frac{3 x^{3}+9 x^{2}+10 x+2}{(x-1)(x+1)^{3}} d x=
$$

We decompose the proper rational fraction into partial fractions using the method of undetermined coefficients:

$$
\begin{aligned}
& \frac{3 x^{3}+9 x^{2}+10 x+2}{(x-1)(x+1)^{3}}=\frac{A}{x-1}+\frac{B_{1}}{x+1}+\frac{B_{2}}{(x+1)^{2}}+\frac{B_{3}}{(x+1)^{3}}= \\
& =\frac{A(x+1)^{3}+B_{1}(x-1)(x+1)^{2}+B_{2}(x-1)(x+1)+B_{3}(x-1)}{(x-1)(x+1)^{3}}= \\
& =\frac{A\left(x^{3}+3 x^{2}+3 x+1\right)+B_{1}\left(x^{3}+x^{2}-x-1\right)+B_{2}\left(x^{2}-1\right)+B_{3}(x-1)}{(x-1)(x+1)^{3}}= \\
& =\frac{\left(A+B_{1}\right) x^{3}+\left(3 A+B_{1}+B_{2}\right) x^{2}+\left(3 A-B_{1}+B_{3}\right) x+\left(A-B_{1}-B_{2}-B_{3}\right)}{(x-1)(x+1)^{3}} \\
& \left\{\begin{array}{l}
A+B_{1}=3 \\
3 A+B_{1}+B_{2}=9 \\
3 A-B_{1}+B_{3}=10 \\
A-B_{1}-B_{2}-B_{3}=2
\end{array}\right.
\end{aligned}
$$

Add the second equation to the fourth:

$$
\left\{\begin{array}{l}
A+B_{1}=3 \\
3 A+B_{1}+B_{2}=9 \\
3 A-B_{1}+B_{3}=10 \\
4 A-B_{3}=11
\end{array}\right.
$$

Add the first equation to the third:

$$
\left\{\begin{array}{l}
A+B_{1}=3 \\
3 A+B_{1}+B_{2}=9 \\
4 A+B_{3}=13 \\
4 A-B_{3}=11
\end{array}\right.
$$

Add the third equation to the fourth:

$$
\begin{aligned}
& \left\{\begin{array}{l}
A+B_{1}=3 \\
3 A+B_{1}+B_{2}=9 \\
4 A+B_{3}=13 \\
8 A=24
\end{array}\right. \\
& \left\{\begin{array}{l}
B_{1}=0 \\
B_{1}+B_{2}=0 \\
B_{3}=1 \\
A=3
\end{array}\right. \\
& \left\{\begin{array}{l}
B_{1}=0 \\
B_{2}=0 \\
B_{3}=1 \\
A=3
\end{array}\right. \\
& \frac{3 x^{3}+9 x^{2}+10 x+2}{(x-1)(x+1)^{3}}=\frac{3}{x-1}+\frac{1}{(x+1)^{3}}
\end{aligned}
$$

Then:

$$
\int \frac{3 x^{3}+9 x^{2}+10 x+2}{(x-1)(x+1)^{3}} d x=\int\left(\frac{3}{x-1}+\frac{1}{(x+1)^{3}}\right) d x=3 \cdot \ln |x-1|-\frac{1}{2(x+1)^{2}}+C
$$

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 $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+6-15$ »

Categories: Kuznetsov's Integral Problems Problem $6 \mid$ Integrals

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## Problem Kuznetsov Integrals 6-16

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