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1

Solved example of base change formula of logarithms

$\int\frac{2x-1}{\left(x+1\right)\left(x-6\right)}dx$
2

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

$\frac{2x-1}{\left(x+1\right)\left(x-6\right)}=\frac{A}{x+1}+\frac{B}{x-6}$
3

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

$2x-1=\left(x+1\right)\left(x-6\right)\left(\frac{A}{x+1}+\frac{B}{x-6}\right)$
4

Multiplying polynomials

$2x-1=\frac{A\left(x+1\right)\left(x-6\right)}{x+1}+\frac{B\left(x+1\right)\left(x-6\right)}{x-6}$
5

Simplifying

$2x-1=A\left(x-6\right)+B\left(x+1\right)$
6

Expand the polynomial

$2x-1=Ax-6A+Bx+B$
7

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}-3=-7A&\:\:\:\:\:\:\:(x=-1) \\ 1=2B-5A&\:\:\:\:\:\:\:(x=1)\end{matrix}$
8

Proceed to solve the system of linear equations

$\begin{matrix} -7A & + & 0B & =-3 \\ -5A & + & 2B & =1\end{matrix}$
9

Rewrite as a coefficient matrix

$\left(\begin{matrix}-7 & 0 & -3 \\ -5 & 2 & 1\end{matrix}\right)$
10

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & \frac{3}{7} \\ 0 & 1 & \frac{11}{7}\end{matrix}\right)$
11

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

$\int\left(\frac{\frac{3}{7}}{x+1}+\frac{\frac{11}{7}}{x-6}\right)dx$
12

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

$\int\frac{\frac{3}{7}}{x+1}dx+\int\frac{\frac{11}{7}}{x-6}dx$
13

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

$\frac{3}{7}\ln\left|x+1\right|+\int\frac{\frac{11}{7}}{x-6}dx$
14

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

$\frac{3}{7}\ln\left|x+1\right|+\frac{11}{7}\ln\left|x-6\right|$
15

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

$\frac{3}{7}\ln\left|x+1\right|+\frac{11}{7}\ln\left|x-6\right|+C_0$

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