Step-by-step Solution

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

Go!
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Final Answer

$\ln\left(\sqrt{x^2-1}\right)+C_0$

Step-by-step Solution

Problem to solve:

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

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

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

Differentiate both sides of the equation $x=\sec\left(\theta \right)$

$dx=\frac{d}{d\theta}\left(\sec\left(\theta \right)\right)$

Find the derivative

$\frac{d}{d\theta}\left(\sec\left(\theta \right)\right)$

Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$

$\sec\left(\theta \right)\tan\left(\theta \right)$
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 trigonometric 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

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

Differentiate both sides of the equation $u=\tan\left(\theta \right)$

$du=\frac{d}{d\theta}\left(\tan\left(\theta \right)\right)$

Find the derivative

$\frac{d}{d\theta}\left(\tan\left(\theta \right)\right)$

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

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

The derivative of the linear function is equal to $1$

$\sec\left(\theta \right)^2$
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

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

Simplify the fraction $\frac{\frac{\sec\left(\theta \right)^2}{u}}{\sec\left(\theta \right)^2}$ by $\sec\left(\theta \right)^2$

$\int\frac{1}{u}du$
9

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

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

$1\ln\left(u\right)$

Any expression multiplied by $1$ is equal to itself

$\ln\left(u\right)$
10

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

$\ln\left(u\right)$

$\ln\left(\tan\left(\theta \right)\right)$
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 $C$

$\ln\left(\sqrt{x^2-1}\right)+C_0$

Final Answer

$\ln\left(\sqrt{x^2-1}\right)+C_0$