Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration method of trigonometric substitution using the substitution
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
Substituting in the original integral, we get
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
Simplify the fraction by $\tan\left(\theta \right)$
Rewrite the trigonometric expression $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ inside the integral
The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
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
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
Isolate $d\theta$ in the previous equation
Substituting $u$ and $d\theta$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Simplify the expression inside the integral
The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$
Replace $u$ with the value that we assigned to it in the beginning: $2\theta $
Simplify $\csc\left(2\theta \right)+\cot\left(2\theta \right)$ using trigonometric identities
Express the variable $\theta$ in terms of the original variable $x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$