# Step-by-step Solution

## Integral of $\frac{x}{x^2-1}$ with respect to x

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### Videos

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$

## Step-by-step explanation

Problem to solve:

$\int\frac{x}{x^2-1}dx$

Choose the solving method

1

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

$u=x^2-1$
2

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

$du=2xdx$
3

Isolate $dx$ in the previous equation

$\frac{du}{2x}=dx$
4

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

$\frac{1}{2}\int\frac{1}{u}du$
5

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

$\frac{1}{2}\cdot 1\ln\left|u\right|$
6

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

$\frac{1}{2}\ln\left|u\right|$
7

Replace $u$ with the value that we assigned to it in the beginning: $x^2-1$

$\frac{1}{2}\ln\left|x^2-1\right|$
8

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

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$

$\frac{1}{2}\ln\left|x^2-1\right|+C_0$
$\int\frac{x}{x^2-1}dx$