Final Answer
Step-by-step Solution
Problem to solve:
Choose the solving method
Rewrite the fraction $\frac{x}{x^2-1}$ inside the integral as the product of two functions: $x\frac{1}{x^2-1}$
We can solve the integral $\int\frac{x}{x^2-1}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the linear function is equal to $1$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
Applying the power of a power property
Factor the difference of squares $x^2-1$ as the product of two conjugated binomials
Rewrite the fraction $\frac{1}{\left(x+1\right)\left(x-1\right)}$ in $2$ simpler fractions using partial fraction decomposition
Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $\left(x+1\right)\left(x-1\right)$
Multiplying polynomials
Simplifying
Expand the polynomial
Assigning values to $x$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{1}{\left(x+1\right)\left(x-1\right)}$ in decomposed fraction equals
We can solve the integral $\int\frac{-\frac{1}{2}}{x+1}dx$ 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 $x+1$ 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 $dx$ 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
Substituting $u$ and $dx$ in the integral and simplify
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $u$ with the value that we assigned to it in the beginning: $x+1$
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left(x+b\right)$, where $b=1$ and $n=-\frac{1}{2}$
We can solve the integral $\int\frac{\frac{1}{2}}{x-1}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting $u$ and $dx$ in the integral and simplify
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $u$ with the value that we assigned to it in the beginning: $x-1$
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left(x+b\right)$, where $b=-1$ and $n=\frac{1}{2}$
Any expression multiplied by $1$ is equal to itself
Expand the integral $\int\left(-\frac{1}{2}\ln\left(x+1\right)+\frac{1}{2}\ln\left(x-1\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $x$ by each term of the polynomial $\left(-\frac{1}{2}\ln\left(x+1\right)+\frac{1}{2}\ln\left(x-1\right)\right)$
We can solve the integral $\int\ln\left(x+1\right)dx$ 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 $x+1$ 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 $dx$ 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
Substituting $u$ and $dx$ in the integral and simplify
The integral of the natural logarithm is given by the following formula, $\displaystyle\int\ln(x)dx=x\ln(x)-x$
Replace $u$ with the value that we assigned to it in the beginning: $x+1$
Solve the product $-(x+1)$
The integral $\int\ln\left(x+1\right)dx$ results in $\left(x+1\right)\ln\left(x+1\right)-\left(x+1\right)$
We can solve the integral $\int\ln\left(x-1\right)dx$ 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 $x-1$ 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 $dx$ 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
Substituting $u$ and $dx$ in the integral and simplify
The integral of the natural logarithm is given by the following formula, $\displaystyle\int\ln(x)dx=x\ln(x)-x$
Replace $u$ with the value that we assigned to it in the beginning: $x-1$
Solve the product $-(x-1)$
The integral $\int\ln\left(x-1\right)dx$ results in $\left(x-1\right)\ln\left(x-1\right)-\left(x-1\right)$
Solve the product $-(x+1)$
Solve the product $-(x-1)$
Simplifying
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$