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We can factor the polynomial $x^6-2x^4+x^2$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $0$
Next, list all divisors of the leading coefficient $a_n$, which equals $1$
The possible roots $\pm\frac{p}{q}$ of the polynomial $x^6-2x^4+x^2$ will then be
We can factor the polynomial $x^6-2x^4+x^2$ using synthetic division (Ruffini's rule). We found that $1$ is a root of the polynomial
Now, divide the polynomial by the root we found $\left(x-1\right)$ using synthetic division (Ruffini's rule). First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $1$. Add the result to the second coefficient and then multiply this by $1$ and so on
In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $\left(x-1\right)$
We can factor the polynomial $\left(x^{5}+x^{4}-x^{3}-x^{2}\right)$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $0$
Next, list all divisors of the leading coefficient $a_n$, which equals $1$
The possible roots $\pm\frac{p}{q}$ of the polynomial $\left(x^{5}+x^{4}-x^{3}-x^{2}\right)$ will then be
We can factor the polynomial $\left(x^{5}+x^{4}-x^{3}-x^{2}\right)$ using synthetic division (Ruffini's rule). We found that $1$ is a root of the polynomial
Now, divide the polynomial by the root we found $\left(x-1\right)$ using synthetic division (Ruffini's rule). First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $1$. Add the result to the second coefficient and then multiply this by $1$ and so on
In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $\left(x-1\right)$
We can factor the polynomial $\left(x^{4}+2x^{3}+x^{2}\right)$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $0$
Next, list all divisors of the leading coefficient $a_n$, which equals $1$
The possible roots $\pm\frac{p}{q}$ of the polynomial $\left(x^{4}+2x^{3}+x^{2}\right)$ will then be
We can factor the polynomial $\left(x^{4}+2x^{3}+x^{2}\right)$ using synthetic division (Ruffini's rule). We found that $-1$ is a root of the polynomial
Now, divide the polynomial by the root we found $\left(x+1\right)$ using synthetic division (Ruffini's rule). First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on
In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $\left(x+1\right)$
Factor the polynomial $\left(x^{3}+x^{2}\right)$ by it's greatest common factor (GCF): $x^2$
Rewrite the expression $\frac{x+8}{x^6-2x^4+x^2}$ inside the integral in factored form
Rewrite the fraction $\frac{x+8}{\left(x-1\right)^2\left(x+1\right)^2x^2}$ in $6$ simpler fractions using partial fraction decomposition
Find the values for the unknown coefficients: $A, B, C, D, F, G$. The first step is to multiply both sides of the equation from the previous step by $\left(x-1\right)^2\left(x+1\right)^2x^2$
Multiply both sides of the equality by $1$ to simplify the fractions
Multiplying polynomials
Simplifying
Assigning values to $x$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{x+8}{\left(x-1\right)^2\left(x+1\right)^2x^2}$ in decomposed fraction equals
Expand the integral $\int\left(\frac{9}{4\left(x-1\right)^2}+\frac{7}{4\left(x+1\right)^2}+\frac{32}{37x^2}+\frac{-2.943243}{x-1}+\frac{2.714414}{x+1}+\frac{\frac{49}{370}}{x}\right)dx$ into $6$ integrals using the sum rule for integrals, to then solve each integral separately
Take the constant $\frac{1}{4}$ out of the integral
Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(c-1\right)\left(x+a\right)^{\left(c-1\right)}}+C$, where $a=-1$, $c=2$ and $n=9$
Simplify the expression inside the integral
The integral $\int\frac{9}{4\left(x-1\right)^2}dx$ results in: $\frac{-9}{4\left(x-1\right)}$
Take the constant $\frac{1}{4}$ out of the integral
Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(c-1\right)\left(x+a\right)^{\left(c-1\right)}}+C$, where $a=1$, $c=2$ and $n=7$
Simplify the expression inside the integral
The integral $\int\frac{7}{4\left(x+1\right)^2}dx$ results in: $\frac{-7}{4\left(x+1\right)}$
Take the constant $\frac{1}{37}$ out of the integral
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
The integral of a function times a constant ($32$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Any expression to the power of $1$ is equal to that same expression
The integral $\int\frac{32}{37x^2}dx$ results in: $\frac{-32}{37x}$
The integral of a function times a constant ($-2.943243$) is equal to the constant times the integral of the function
Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=-1$ and $n=1$
The integral $\int\frac{-2.943243}{x-1}dx$ results in: $-2.943243\ln\left(x-1\right)$
The integral of a function times a constant ($2.714414$) is equal to the constant times the integral of the function
Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=1$ and $n=1$
The integral $\int\frac{2.714414}{x+1}dx$ results in: $2.714414\ln\left(x+1\right)$
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
The integral $\int\frac{\frac{49}{370}}{x}dx$ results in: $\frac{49}{370}\ln\left(x\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$