** Final answer to the problem

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** Step-by-step Solution **

** How should I solve this problem?

- Integrate by trigonometric substitution
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
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We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration method of trigonometric substitution using the substitution

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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

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Substituting in the original integral, we get

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

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

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Rewrite the trigonometric expression $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ inside the integral

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Reduce $\sec\left(\theta \right)\csc\left(\theta \right)$ by applying trigonometric identities

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Rewrite the trigonometric expression $\frac{\csc\left(\theta \right)}{\cos\left(\theta \right)}$ inside the integral

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The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

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

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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

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Isolate $d\theta$ in the previous equation

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

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

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Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\int\csc\left(u\right)du$

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The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

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Replace $u$ with the value that we assigned to it in the beginning: $2\theta $

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Simplify $\csc\left(2\theta \right)+\cot\left(2\theta \right)$ using trigonometric identities

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

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

** Final answer to the problem

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