Final answer to the problem
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
Differentiate both sides of the equation $x=\sec\left(\theta \right)$
Find the derivative
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
The derivative of the linear function is equal to $1$
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
Multiplying the fraction by $\sec\left(\theta \right)\tan\left(\theta \right)$
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 $\tan\left(\theta \right)$
Simplify $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ by applying trigonometric identities
Rewrite the trigonometric expression $\frac{\sec\left(\theta \right)^2}{\tan\left(\theta \right)}$ inside the integral
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Multiplying the fraction by $\csc\left(\theta \right)$
Any expression multiplied by $1$ is equal to itself
Reduce $\sec\left(\theta \right)\csc\left(\theta \right)$ by applying trigonometric identities
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Divide fractions $\frac{\frac{1}{\sin\left(\theta \right)}}{\cos\left(\theta \right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Simplify $\sin\left(\theta \right)\cos\left(\theta \right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$
Divide fractions $\frac{1}{\frac{\sin\left(2\theta \right)}{2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Rewrite the trigonometric expression $\frac{\csc\left(\theta \right)}{\cos\left(\theta \right)}$ 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
Differentiate both sides of the equation $u=2\theta $
Find the derivative
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
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
Rearrange the equation
Divide both sides of the equation by $2$
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
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\int\csc\left(u\right)du$
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 $
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
Simplify the logarithm $\ln\left(\frac{1}{\sqrt{x^2-1}}\right)$
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Express the variable $\theta$ in terms of the original variable $x$
Any expression multiplied by $1$ is equal to itself
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
