Integral of (2x+8)/(9x+20+x^2)

\int\frac{2x+8}{x^2+9x+20}dx

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Answer

$2\ln\left|5+x\right|+C_0$

Step by step solution

Problem

$\int\frac{2x+8}{x^2+9x+20}dx$
1

Factor the trinomial $20+9x+x^2$ finding two numbers that multiply to form $20$ and added form $9$

$\begin{matrix}\left(4\right)\left(5\right)=20\\ \left(4\right)+\left(5\right)=9\end{matrix}$
2

Thus

$\int\frac{8+2x}{\left(5+x\right)\left(4+x\right)}dx$
3

Using partial fraction decomposition, the fraction $\frac{8+2x}{\left(5+x\right)\left(4+x\right)}$ can be rewritten as

$\frac{8+2x}{\left(5+x\right)\left(4+x\right)}=\frac{A}{5+x}+\frac{B}{4+x}$
4

Now we need to find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(5+x\right)\left(4+x\right)$

$8+2x=\left(\frac{A}{5+x}+\frac{B}{4+x}\right)\left(5+x\right)\left(4+x\right)$
5

Multiplying polynomials

$8+2x=\frac{A\left(5+x\right)\left(4+x\right)}{5+x}+\frac{B\left(5+x\right)\left(4+x\right)}{4+x}$
6

Simplifying

$8+2x=A\left(4+x\right)+B\left(5+x\right)$
7

Expand the polynomial

$8+2x=4A+Ax+5B+Bx$
8

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

$\begin{matrix}0=B&\:\:\:\:\:\:\:(x=-4) \\ -2=-A&\:\:\:\:\:\:\:(x=-5)\end{matrix}$
9

Proceed to solve the system of linear equations

$\begin{matrix}0A & + & 1B & =0 \\ -1A & + & 0B & =-2\end{matrix}$
10

Rewrite as a coefficient matrix

$\left(\begin{matrix}0 & 1 & 0 \\ -1 & 0 & -2\end{matrix}\right)$
11

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 2 \\ 0 & 1 & 0\end{matrix}\right)$
12

The decomposed integral equivalent is

$\int\frac{2}{5+x}dx$
13

Solve the integral $\int\frac{2}{5+x}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=5+x \\ du=dx\end{matrix}$
14

Substituting $u$ and $dx$ in the integral

$\int\frac{2}{u}du$
15

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$2\ln\left|u\right|$
16

Substitute $u$ back for it's value, $5+x$

$2\ln\left|5+x\right|$
17

Add the constant of integration

$2\ln\left|5+x\right|+C_0$

Answer

$2\ln\left|5+x\right|+C_0$