1
Solved example of polynomials
$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$
2
Rewrite the fraction $\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}$ in $3$ simpler fractions using partial fraction decomposition
$\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}=\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}$
3
Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)^2\left(x+2\right)$
$5x^2+14x+10=\left(x+1\right)^2\left(x+2\right)\left(\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}\right)$
4
Multiplying polynomials
$5x^2+14x+10=\frac{A\left(x+1\right)^2\left(x+2\right)}{x+2}+\frac{B\left(x+1\right)^2\left(x+2\right)}{\left(x+1\right)^2}+\frac{C\left(x+1\right)^2\left(x+2\right)}{x+1}$
$5x^2+14x+10=A\left(x+1\right)^2+B\left(x+2\right)+C\left(x+2\right)\left(x+1\right)$
$5x^2+14x+10=A\left(x+1\right)^2+Bx+2B+Cx^2+2Cx+Cx+2C$
7
Assigning values to $x$ we obtain the following system of equations
$\begin{matrix}1=B&\:\:\:\:\:\:\:(x=-1) \\ 2=A&\:\:\:\:\:\:\:(x=-2) \\ 29=6C+3B+4A&\:\:\:\:\:\:\:(x=1)\end{matrix}$
8
Proceed to solve the system of linear equations
$\begin{matrix}0A & + & 1B & + & 0C & =1 \\ 1A & + & 0B & + & 0C & =2 \\ 4A & + & 3B & + & 6C & =29\end{matrix}$
9
Rewrite as a coefficient matrix
$\left(\begin{matrix}0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 2 \\ 4 & 3 & 6 & 29\end{matrix}\right)$
10
Reducing the original matrix to a identity matrix using Gaussian Elimination
$\left(\begin{matrix}1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3\end{matrix}\right)$
11
The integral of $\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}$ in decomposed fraction equals
$\int\left(\frac{2}{x+2}+\frac{1}{\left(x+1\right)^2}+\frac{3}{x+1}\right)dx$
12
The integral of the sum of two or more functions is equal to the sum of their integrals
$\int\frac{2}{x+2}dx+\int\frac{1}{\left(x+1\right)^2}dx+\int\frac{3}{x+1}dx$
13
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=2$ and $n=2$
$2\ln\left|x+2\right|+\int\frac{1}{\left(x+1\right)^2}dx+\int\frac{3}{x+1}dx$
14
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=1$ and $n=3$
$2\ln\left|x+2\right|+\int\frac{1}{\left(x+1\right)^2}dx+3\ln\left|x+1\right|$
15
Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(x+a\right)^{\left(c-1\right)}\left(c-1\right)}$, where $a=1$, $c=2$ and $n=1$
$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|$
16
As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration
$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$