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1

Solved example of gaussian elimination

$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$
2

Rewrite the fraction $\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}=\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}$
3

Find the values for the unknown coefficients: $A, B, C$. The first step is to multiply both sides of the equation from the previous step by $\left(x+2\right)\left(x+1\right)^2$

$5x^2+14x+10=\left(x+2\right)\left(x+1\right)^2\left(\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}\right)$
4

Multiplying polynomials

$5x^2+14x+10=\frac{A\left(x+2\right)\left(x+1\right)^2}{x+2}+\frac{B\left(x+2\right)\left(x+1\right)^2}{\left(x+1\right)^2}+\frac{C\left(x+2\right)\left(x+1\right)^2}{x+1}$
5

Simplifying

$5x^2+14x+10=A\left(x+1\right)^2+B\left(x+2\right)+C\left(x+2\right)\left(x+1\right)$
6

Expand the polynomial

$5x^2+14x+10=A\left(x+1\right)^2+Bx+2B+x^2C+2xC+Cx+2C$
7

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}2=A&\:\:\:\:\:\:\:(x=-2) \\ 58=12C+4B+9A&\:\:\:\:\:\:\:(x=2) \\ 1=B&\:\:\:\:\:\:\:(x=-1)\end{matrix}$
8

Proceed to solve the system of linear equations

$\begin{matrix}1A & + & 0B & + & 0C & =2 \\ 9A & + & 4B & + & 12C & =58 \\ 0A & + & 1B & + & 0C & =1\end{matrix}$
9

Rewrite as a coefficient matrix

$\left(\begin{matrix}1 & 0 & 0 & 2 \\ 9 & 4 & 12 & 58 \\ 0 & 1 & 0 & 1\end{matrix}\right)$
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 $\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}$ in decomposed fraction equals

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

The integral of the sum of two or more functions is equal to the sum of their integrals

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

The integral of the sum of two or more functions is equal to the sum of their integrals

$\int\frac{2}{x+2}dx+\int\frac{1}{\left(x+1\right)^2}dx+\int\frac{3}{x+1}dx$
12

Simplifying

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

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$2\int\frac{1}{2+x}dx$

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=2$ and $n=1$

$2\ln\left|x+2\right|$
13

The integral $\int\frac{2}{x+2}dx$ results in: $2\ln\left|x+2\right|$

$2\ln\left|x+2\right|$

We can solve the integral $\int\frac{1}{\left(x+1\right)^2}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a part of the integral with a new variable, which when substituted makes the integral easier. We see that $x+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x+1$

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=dx$

Substituting $u$ and $dx$ in the integral and simplify

$2\ln\left|x+2\right|+\int\frac{1}{u^2}du+\int\frac{3}{x+1}dx$

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int u^{-2}du$

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$-u^{-1}$

Substitute $u$ back with the value that we assigned to it: $x+1$

$-\left(x+1\right)^{-1}$

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{-1}{x+1}$
14

The integral $\int\frac{1}{\left(x+1\right)^2}dx$ results in: $\frac{-1}{x+1}$

$\frac{-1}{x+1}$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$3\int\frac{1}{1+x}dx$

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=1$ and $n=1$

$3\ln\left|x+1\right|$
15

The integral $\int\frac{3}{x+1}dx$ results in: $3\ln\left|x+1\right|$

$3\ln\left|x+1\right|$
16

Gather the results of all integrals

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|$
17

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$

Final Answer

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$

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