Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Weierstrass Substitution
- Produit de binômes avec terme commun
- Méthode FOIL
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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
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$$=\tan\left(\theta \right)^2$, where $x=\theta $
Apply the formula: $\frac{a}{a^n}$$=\frac{1}{a^{\left(n-1\right)}}$, where $a=\tan\left(\theta \right)$ and $n=2$
Rewrite the trigonometric expression $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ inside the integral
Reduce $\sec\left(\theta \right)\csc\left(\theta \right)$ by applying trigonometric identities
Rewrite the trigonometric expression $\frac{\csc\left(\theta \right)}{\cos\left(\theta \right)}$ inside the integral
Apply the formula: $\int cxdx$$=c\int xdx$, where $c=2$ and $x=\csc\left(2\theta \right)$
We can solve the integral $\int\csc\left(2\theta \right)d\theta$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2\theta $ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $d\theta$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $d\theta$ in the previous equation
Substituting $u$ and $d\theta$ in the integral and simplify
Apply the formula: $\int\frac{x}{c}dx$$=\frac{1}{c}\int xdx$, where $c=2$ and $x=\csc\left(u\right)$
Apply the formula: $\frac{a}{b}c$$=\frac{ca}{b}$, where $a=1$, $b=2$, $c=2$, $a/b=\frac{1}{2}$ and $ca/b=2\cdot \left(\frac{1}{2}\right)\int\csc\left(u\right)du$
Apply the formula: $\int\csc\left(\theta \right)dx$$=-\ln\left(\csc\left(\theta \right)+\cot\left(\theta \right)\right)+C$, where $x=u$
Replace $u$ with the value that we assigned to it in the beginning: $2\theta $
Apply the trigonometric identity: $\csc\left(nx\right)+\cot\left(nx\right)$$=\cot\left(\frac{n}{2}x\right)$, where $x=\theta $, $nx=2\theta $ and $n=2$
Express the variable $\theta$ in terms of the original variable $x$
Apply the formula: $1x$$=x$, where $x=\frac{1}{2}\ln\left(x^2-1\right)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
