** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Calculate the power $\sqrt{1}$

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Calculate the power $\sqrt{1}$

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Factor the difference of squares $x^2-1$ as the product of two conjugated binomials

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Rewrite the fraction $\frac{x}{\left(x+1\right)\left(x-1\right)}$ in $2$ simpler fractions using partial fraction decomposition

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

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Multiplying polynomials

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Simplifying

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Expand the polynomial

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

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Proceed to solve the system of linear equations

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Rewrite as a coefficient matrix

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Reducing the original matrix to a identity matrix using Gaussian Elimination

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The integral of $\frac{x}{\left(x+1\right)\left(x-1\right)}$ in decomposed fraction equals

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Expand the integral $\int\left(\frac{1}{2\left(x+1\right)}+\frac{1}{2\left(x-1\right)}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

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We can solve the integral $\int\frac{1}{2\left(x+1\right)}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), 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

Differentiate both sides of the equation $u=x+1$

Find the derivative

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($1$) is equal to zero

The derivative of the linear function is equal to $1$

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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

Take the constant $\frac{1}{2}$ out of the integral

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

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

Replace $u$ with the value that we assigned to it in the beginning: $x+1$

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The integral $\frac{1}{2}\int\frac{1}{u}du$ results in: $\frac{1}{2}\ln\left(x+1\right)$

Take the constant $\frac{1}{2}$ out of the integral

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

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

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Gather the results of all integrals

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

** Final Answer

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