Step-by-step Solution

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

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Final Answer

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

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 method of trigonometric substitution using the substitution

$x=\sec\left(\theta \right)$
2

Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above

$dx=\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
3

Substituting in the original integral, we get

$\int\frac{\sec\left(\theta \right)^2\tan\left(\theta \right)}{\sec\left(\theta \right)^2-1}d\theta$
4

Apply the identity: $\sec\left(x\right)^2-1$$=\tan\left(x\right)^2$, where $x=\theta $

$\int\frac{\sec\left(\theta \right)^2\tan\left(\theta \right)}{\tan\left(\theta \right)^2}d\theta$
5

Simplify the fraction by $\tan\left(\theta \right)$

$\int\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}d\theta$
6

We can solve the integral $\int\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}d\theta$ 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 $\tan\left(\theta \right)$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=\tan\left(\theta \right)$
7

Now, in order to rewrite $d\theta$ 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

$du=\sec\left(\theta \right)^2d\theta$
8

Isolate $d\theta$ in the previous equation

$\frac{du}{\sec\left(\theta \right)^2}=d\theta$
9

Substituting $u$ and $d\theta$ in the integral and simplify

$\int\frac{1}{u}du$
10

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

$\ln\left|u\right|$
11

Replace $u$ with the value that we assigned to it in the beginning: $\tan\left(\theta \right)$

$\ln\left|\tan\left(\theta \right)\right|$
12

Express the variable $\theta$ in terms of the original variable $x$

$\ln\left|\frac{\sqrt{x^2-1}}{1}\right|$
13

Any expression divided by one ($1$) is equal to that same expression

$\ln\left|\sqrt{x^2-1}\right|$
14

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

$\ln\left|\sqrt{x^2-1}\right|+C_0$
15

Simplify $\ln\left|\sqrt{x^2-1}\right|$ by applying logarithm properties

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

Final Answer

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