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### Difficult Problems

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

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

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

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

$\begin{matrix}1=B&\:\:\:\:\:\:\:(x=-1) \\ 2=A&\:\:\:\:\:\:\:(x=-2) \\ 29=4A+B+2B+2C+4C&\:\:\:\:\:\:\:(x=1)\end{matrix}$
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

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

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