Integral of (14x+10+5x^2)/((x+2)(x+1)^2)

\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx

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Answer

$-53\frac{-\frac{29}{5}}{1+x}\ln\left|1+x\right|-614.8\ln\left|2+x\right|\ln\left|1+x\right|+561.8\ln\left|1+x\right|^2+C_0$

Step by step solution

Problem

$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$
1

Use the complete the square method to factor the trinomial of the form $ax^2+bx+c$. Take common factor $a$ ($5$) to all terms

$\int\frac{5\left(2+\frac{14}{5}x+x^2\right)}{\left(1+x\right)^2\left(2+x\right)}dx$
2

Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$

$\int\frac{5\left(-\frac{49}{25}+\frac{49}{25}+2+\frac{14}{5}x+x^2\right)}{\left(1+x\right)^2\left(2+x\right)}dx$
3

Factor the perfect square trinomial $x^2+\frac{14}{5}x+\frac{49}{25}$

$\int\frac{5\left(-\frac{49}{25}+2+\left(x-\frac{7}{5}\right)^2\right)}{\left(1+x\right)^2\left(2+x\right)}dx$
4

Subtract the values $2$ and $-\frac{49}{25}$

$\int\frac{5\left(\left(x-\frac{7}{5}\right)^2+\frac{1}{25}\right)}{\left(1+x\right)^2\left(2+x\right)}dx$
5

Taking the constant out of the integral

$5\int\frac{\left(x-\frac{7}{5}\right)^2+\frac{1}{25}}{\left(1+x\right)^2\left(2+x\right)}dx$
6

Using partial fraction decomposition, the fraction $\frac{\left(x-\frac{7}{5}\right)^2+\frac{1}{25}}{\left(1+x\right)^2\left(2+x\right)}$ can be rewritten as

$\frac{\left(x-\frac{7}{5}\right)^2+\frac{1}{25}}{\left(1+x\right)^2\left(2+x\right)}=\frac{A}{\left(1+x\right)^2}+\frac{B}{2+x}+\frac{C}{1+x}$
7

Now we need to find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(1+x\right)^2\left(2+x\right)$

$\left(x-\frac{7}{5}\right)^2+\frac{1}{25}=\left(\frac{A}{\left(1+x\right)^2}+\frac{B}{2+x}+\frac{C}{1+x}\right)\left(1+x\right)^2\left(2+x\right)$
8

Multiplying polynomials

$\left(x-\frac{7}{5}\right)^2+\frac{1}{25}=\frac{A\left(1+x\right)^2\left(2+x\right)}{\left(1+x\right)^2}+\frac{B\left(1+x\right)^2\left(2+x\right)}{2+x}+\frac{C\left(1+x\right)^2\left(2+x\right)}{1+x}$
9

Simplifying

$\left(x-\frac{7}{5}\right)^2+\frac{1}{25}=A\left(2+x\right)+B\left(1+x\right)^2+C\left(1+x\right)\left(2+x\right)$
10

Expand the polynomial

$\left(x-\frac{7}{5}\right)^2+\frac{1}{25}=B\left(1+x\right)^2+2A+Ax+2C+2Cx+Cx+Cx^2$
11

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

$\begin{matrix}\frac{29}{5}=A&\:\:\:\:\:\:\:(x=-1) \\ \frac{1}{5}=4B+2A+A+4C+2C&\:\:\:\:\:\:\:(x=1) \\ \frac{58}{5}=B&\:\:\:\:\:\:\:(x=-2)\end{matrix}$
12

Proceed to solve the system of linear equations

$\begin{matrix}1A & + & 0B & + & 0C & =\frac{29}{5} \\ 3A & + & 4B & + & 6C & =\frac{1}{5} \\ 0A & + & 1B & + & 0C & =\frac{58}{5}\end{matrix}$
13

Rewrite as a coefficient matrix

$\left(\begin{matrix}1 & 0 & 0 & \frac{29}{5} \\ 3 & 4 & 6 & \frac{1}{5} \\ 0 & 1 & 0 & \frac{58}{5}\end{matrix}\right)$
14

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 0 & \frac{29}{5} \\ 0 & 1 & 0 & \frac{58}{5} \\ 0 & 0 & 1 & -\frac{53}{5}\end{matrix}\right)$
15

The decomposed integral equivalent is

$5\int\left(\frac{\frac{29}{5}}{\left(1+x\right)^2}+\frac{\frac{58}{5}}{2+x}+\frac{-\frac{53}{5}}{1+x}\right)dx$
16

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

$5\left(\int\frac{\frac{29}{5}}{\left(1+x\right)^2}dx+\int\frac{\frac{58}{5}}{2+x}dx+\int\frac{-\frac{53}{5}}{1+x}dx\right)$
17

Apply the formula: $\int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|$, where $b=2$ and $n=\frac{58}{5}$

$5\left(\int\frac{\frac{29}{5}}{\left(1+x\right)^2}dx+\frac{58}{5}\ln\left|2+x\right|+\int\frac{-\frac{53}{5}}{1+x}dx\right)$
18

Apply the formula: $\int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|$, where $b=1$ and $n=-\frac{53}{5}$

$5\left(\int\frac{\frac{29}{5}}{\left(1+x\right)^2}dx+\frac{58}{5}\ln\left|2+x\right|-\frac{53}{5}\ln\left|1+x\right|\right)$
19

Apply the formula: $\int\frac{n}{\left(a+x\right)^2}dx$$=\frac{-n}{a+x}$, where $a=1$ and $n=\frac{29}{5}$

$5\left(\frac{-\frac{29}{5}}{1+x}+\frac{58}{5}\ln\left|2+x\right|-\frac{53}{5}\ln\left|1+x\right|\right)$
20

Multiplying polynomials $-53\ln\left|1+x\right|$ and $-\frac{53}{5}\ln\left|1+x\right|+\frac{58}{5}\ln\left|2+x\right|$

$-53\frac{-\frac{29}{5}}{1+x}\ln\left|1+x\right|-614.8\ln\left|2+x\right|\ln\left|1+x\right|+561.8\ln\left|1+x\right|^2$
21

Add the constant of integration

$-53\frac{-\frac{29}{5}}{1+x}\ln\left|1+x\right|-614.8\ln\left|2+x\right|\ln\left|1+x\right|+561.8\ln\left|1+x\right|^2+C_0$

Answer

$-53\frac{-\frac{29}{5}}{1+x}\ln\left|1+x\right|-614.8\ln\left|2+x\right|\ln\left|1+x\right|+561.8\ln\left|1+x\right|^2+C_0$

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