## Final Answer

## Step-by-step explanation

Problem to solve:

Choose the solving method

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

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

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 $\left(x+1\right)\left(x-1\right)$

Multiplying polynomials

Simplifying

Expand the polynomial

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

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

We can solve the integral $\int\frac{\frac{1}{2}}{x+1}dx$ 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

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

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

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

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

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 $C$