** Final Answer

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** Step-by-step Solution **

** Specify the solving method

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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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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

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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

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

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

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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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