## Problem Statement

Find the indefinite integral:

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

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

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

$$
\begin{aligned}
& \frac{x^{3}+4 x^{2}+3 x+2}{(x+1)^{2}\left(x^{2}+1\right)}=\frac{A_{1}}{x+1}+\frac{A_{2}}{(x+1)^{2}}+\frac{B x+C}{x^{2}+1}= \\
& =\frac{A_{1}(x+1)\left(x^{2}+1\right)+A_{2}\left(x^{2}+1\right)+(B x+C)(x+1)^{2}}{(x+1)^{2}\left(x^{2}+1\right)}= \\
& =\frac{A_{1}\left(x^{3}+x^{2}+x+1\right)+A_{2}\left(x^{2}+1\right)+B\left(x^{3}+2 x^{2}+x\right)+C\left(x^{2}+2 x+1\right)}{(x+1)^{2}\left(x^{2}+1\right)}= \\
& =\frac{\left(A_{1}+B\right) x^{3}+\left(A_{1}+A_{2}+2 B+C\right) x^{2}+\left(A_{1}+B+2 C\right) x+\left(A_{1}+A_{2}+C\right)}{(x+1)^{2}\left(x^{2}+1\right)} \\
& \left\{\begin{array}{l}
A_{1}+B=1 \\
A_{1}+A_{2}+2 B+C=4 \\
A_{1}+B+2 C=3 \\
A_{1}+A_{2}+C=2
\end{array}\right.
\end{aligned}
$$

Subtract the fourth equation from the second:

$$
\left\{\begin{array}{l}
A_{1}+B=1 \\
2 B=2 \\
A_{1}+B+2 C=3 \\
A_{1}+A_{2}+C=2
\end{array}\right.
$$

$$
\begin{aligned}
& \left\{\begin{array}{l}
A_{1}=0 \\
B=1 \\
A_{1}+2 C=2 \\
A_{1}+A_{2}+C=2
\end{array}\right. \\
& \left\{\begin{array}{l}
A_{1}=0 \\
B=1 \\
2 C=2
\end{array}\right. \\
& A_{2}+C=2 \\
& \left\{\begin{array}{l}
A_{1}=0 \\
B=1 \\
C=1 \\
A_{2}=1
\end{array}\right. \\
& \frac{x^{3}+4 x^{2}+3 x+2}{(x+1)^{2}\left(x^{2}+1\right)}=\frac{1}{(x+1)^{2}}+\frac{x+1}{x^{2}+1}
\end{aligned}
$$

Then:

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+7-2$ »

Categories: Kuznetsov's Problem Book Integrals Problem 7 | Integrals

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## Problem Kuznetsov Integrals $7-3$

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