Solved example of base change formula of logarithms

Rewrite the fraction $\frac{2x-1}{\left(x+1\right)\left(x-6\right)}$ in $2$ simpler fractions using partial fraction decomposition

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)\left(x-6\right)$

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 $\frac{2x-1}{\left(x+1\right)\left(x-6\right)}$ in decomposed fraction equals

The integral of the sum of two or more functions is equal to the sum of their integrals

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=1$ and $n=\frac{3}{7}$

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=-6$ and $n=\frac{11}{7}$

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration