Final Answer
Step-by-step Solution
Problem to solve:
Specify the solving method
Factor the difference of squares $x^2-1$ as the product of two conjugated binomials
Rewrite the expression $\frac{x}{x^2-1}$ inside the integral in factored form
Rewrite the fraction $\frac{x}{\left(x+1\right)\left(x-1\right)}$ 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 $\frac{x}{\left(x+1\right)\left(x-1\right)}$ in decomposed fraction equals
Expand the integral $\int\left(\frac{1}{2\left(x+1\right)}+\frac{1}{2\left(x-1\right)}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Take the constant $\frac{1}{2}$ out of the integral
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left(x+b\right)+C$, where $b=1$ and $n=1$
The integral $\int\frac{1}{2\left(x+1\right)}dx$ results in: $\frac{1}{2}\ln\left(x+1\right)$
Take the constant $\frac{1}{2}$ out of the integral
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left(x+b\right)+C$, where $b=-1$ and $n=1$
The integral $\int\frac{1}{2\left(x-1\right)}dx$ results in: $\frac{1}{2}\ln\left(x-1\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$