Integral of (2x-1)/((x+1)(x-6))

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

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Answer

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

Step by step solution

Problem

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

Using partial fraction decomposition, the fraction $\frac{2x-1}{\left(x-6\right)\left(1+x\right)}$ can be rewritten as

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

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

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

Multiplying polynomials

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

Simplifying

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

Expand the polynomial

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

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

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

Proceed to solve the system of linear equations

$\begin{matrix}0A & - & 7B & =-3 \\ 2A & - & 5B & =1\end{matrix}$
8

Rewrite as a coefficient matrix

$\left(\begin{matrix}0 & -7 & -3 \\ 2 & -5 & 1\end{matrix}\right)$
9

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & \frac{11}{7} \\ 0 & 1 & \frac{3}{7}\end{matrix}\right)$
10

The decomposed integral equivalent is

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

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

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

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

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

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

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

Add the constant of integration

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

Answer

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

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Problem Analysis

Main topic:

Integrals by partial fraction expansion

Time to solve it:

0.32 seconds

Views:

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