## Final Answer

## Step-by-step Solution

Problem to solve:

Solving method

We can solve the integral $\int\frac{x}{x^2-1}dx$ 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 $x^2-1$ 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 $u=x^2-1$

Find the derivative

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

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

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

Isolate $dx$ in the previous equation

Simplify the fraction $\frac{\frac{x}{u}}{2x}$ by $x$

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

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

Multiply $\frac{1}{2}$ times $1$

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: $x^2-1$

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