## Problem Statement

Find the indefinite integral:

$$
\int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x
$$
## Solution

$$
\int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x=
$$

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

$$
\begin{aligned}
& \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)}=\frac{A x+B}{x^{2}+x+1}+\frac{C x+D}{x^{2}+x+2}= \\
& =\frac{(A x+B)\left(x^{2}+x+2\right)+(C x+D)\left(x^{2}+x+1\right)}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)}= \\
& =\frac{A x^{3}+A x^{2}+2 A x+B x^{2}+B x+2 B+C x^{3}+C x^{2}+C x+D x^{2}+D x+D}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)}= \\
& =\frac{(A+C) x^{3}+(A+B+C+D) x^{2}+(2 A+B+C+D) x+(2 B+D)}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} \\
& \left\{\begin{array}{l}
A+C=2 \\
A+B+C+D=7 \\
2 A+B+C+D=7 \\
2 B+D=9
\end{array}\right.
\end{aligned}
$$

Subtract the second equation from the third:

$$
\begin{aligned}
& \left\{\begin{array}{l}
A+C=2 \\
A+B+C+D=7 \\
A=0 \\
2 B+D=9
\end{array}\right. \\
& \left\{\begin{array}{l}
C=2 \\
B+C+D=7 \\
A=0 \\
2 B+D=9
\end{array}\right.
\end{aligned}
$$

$$
\left\{\begin{array}{l}
C=2 \\
B+D=5 \\
A=0 \\
2 B+D=9
\end{array}\right.
$$

Subtract the second equation from the fourth:

$$
\begin{aligned}
& \left\{\begin{array}{l}
C=2 \\
B+D=5 \\
A=0 \\
B=4
\end{array}\right. \\
& \left\{\begin{array}{l}
C=2 \\
D=1 \\
A=0 \\
B=4
\end{array}\right. \\
& \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)}=\frac{4}{x^{2}+x+1}+\frac{2 x+1}{x^{2}+x+2}
\end{aligned}
$$

Then:

$$
\begin{aligned}
& \int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x=\int\left(\frac{4}{x^{2}+x+1}+\frac{2 x+1}{x^{2}+x+2}\right) d x= \\
& =\int \frac{16}{4 x^{2}+4 x+4} d x+\int \frac{2 x+1}{x^{2}+x+2} d x= \\
& =8 \cdot \int \frac{d(2 x+1)}{(2 x+1)^{2}+3}+\int \frac{d\left(x^{2}+x+2\right)}{x^{2}+x+2} d x= \\
& =\frac{8}{\sqrt{3}} \cdot \operatorname{arctg}\left(\frac{2 x+1}{\sqrt{3}}\right)+\ln \left(x^{2}+x+2\right)+C
\end{aligned}
$$

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 $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+7-20$ »

Categories: Kuznetsov's Integral Problems 7 | Integrals

- Last edited: 15:27, 21 April 2009.
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## Problem Kuznetsov Integrals $7-21$

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