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1

Solved example of power rule

$\int\:\frac{-x^2+8x^2-9x+2}{\left(x^2+1\right)\left(x-3\right)^2}dx$
2

Adding $-1x^2$ and $8x^2$

$\int\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}dx$
3

Rewrite the fraction $\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}=\frac{Ax+B}{x^2+1}+\frac{C}{\left(x-3\right)^2}+\frac{D}{x-3}$
4

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-3\right)^2\left(x^2+1\right)$

$7x^2-9x+2=\left(x-3\right)^2\left(x^2+1\right)\left(\frac{Ax+B}{x^2+1}+\frac{C}{\left(x-3\right)^2}+\frac{D}{x-3}\right)$
5

Multiplying polynomials

$7x^2-9x+2=\frac{\left(x-3\right)^2\left(x^2+1\right)\left(Ax+B\right)}{x^2+1}+\frac{C\left(x-3\right)^2\left(x^2+1\right)}{\left(x-3\right)^2}+\frac{D\left(x-3\right)^2\left(x^2+1\right)}{x-3}$
6

Simplifying

$7x^2-9x+2=\left(x-3\right)^2\left(Ax+B\right)+C\left(x^2+1\right)+D\left(x^2+1\right)\left(x-3\right)$
7

Expand the polynomial

$7x^2-9x+2=A\left(x-3\right)^2x+B\left(x-3\right)^2+Cx^2+C+Dx^{3}+Dx-3Dx^2-3D$
8

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

$\begin{matrix}18=-8D+2C-16A+16B&\:\:\:\:\:\:\:(x=-1) \\ 0=-4D+2C+4A+4B&\:\:\:\:\:\:\:(x=1) \\ 92=-60D+10C-108A+36B&\:\:\:\:\:\:\:(x=-3) \\ 38=10C&\:\:\:\:\:\:\:(x=3)\end{matrix}$
9

Proceed to solve the system of linear equations

$\begin{matrix} -16A & + & 16B & + & 2C & - & 8D & =18 \\ 4A & + & 4B & + & 2C & - & 4D & =0 \\ -108A & + & 36B & + & 10C & - & 60D & =92 \\ 0A & + & 0B & + & 10C & + & 0D & =38\end{matrix}$
10

Rewrite as a coefficient matrix

$\left(\begin{matrix}-16 & 16 & 2 & -8 & 18 \\ 4 & 4 & 2 & -4 & 0 \\ -108 & 36 & 10 & -60 & 92 \\ 0 & 0 & 10 & 0 & 38\end{matrix}\right)$
11

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 0 & 0 & -\frac{51}{50} \\ 0 & 1 & 0 & 0 & \frac{7}{50} \\ 0 & 0 & 1 & 0 & \frac{19}{5} \\ 0 & 0 & 0 & 1 & \frac{51}{50}\end{matrix}\right)$
12

The integral of $\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}$ in decomposed fraction equals

$\int\left(\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}+\frac{\frac{19}{5}}{\left(x-3\right)^2}+\frac{\frac{51}{50}}{x-3}\right)dx$
13

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

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\int\frac{\frac{19}{5}}{\left(x-3\right)^2}dx+\int\frac{\frac{51}{50}}{x-3}dx$
14

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=-3$ and $n=\frac{51}{50}$

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\int\frac{\frac{19}{5}}{\left(x-3\right)^2}dx+\frac{51}{50}\ln\left|x-3\right|$
15

Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(x+a\right)^{\left(c-1\right)}\left(c-1\right)}$, where $a=-3$, $c=2$ and $n=\frac{19}{5}$

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\frac{-\frac{19}{5}}{\left(2-1\right)\left(x-3\right)^{\left(2-1\right)}}+\frac{51}{50}\ln\left|x-3\right|$
16

Subtract the values $2$ and $-1$

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\frac{-\frac{19}{5}}{1\left(x-3\right)^{1}}+\frac{51}{50}\ln\left|x-3\right|$
17

Any expression to the power of $1$ is equal to that same expression

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
18

Split the fraction $\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}$ inside the integral, in two terms with common denominator $x^2+1$

$\int\left(\frac{-\frac{51}{50}x}{x^2+1}+\frac{\frac{7}{50}}{x^2+1}\right)dx+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
19

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

$\int\frac{-\frac{51}{50}x}{x^2+1}dx+\int\frac{\frac{7}{50}}{x^2+1}dx+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
20

Taking the constant out of the integral

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\int\frac{\frac{7}{50}}{x^2+1}dx+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
21

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\int\frac{1}{1+x^2}dx+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
22

Solve the integral applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\left(\frac{1}{\sqrt{1}}\right)arctan\left(\frac{x}{\sqrt{1}}\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
23

The square root of $1$ is $1$

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\left(\frac{1}{1}\right)arctan\left(\frac{x}{1}\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
24

Any expression divided by one is equal to that same expression

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
25

Solve the integral $\int\frac{x}{x^2+1}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=x^2+1 \\ du=2xdx\end{matrix}$
26

Isolate $dx$ in the previous equation

$\frac{du}{2x}=dx$
27

Substituting $u$ and $dx$ in the integral and simplify

$-\frac{51}{50}\int\frac{1}{2u}du+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
28

Take the constant out of the integral

$-\frac{51}{50}\cdot \frac{1}{2}\int\frac{1}{u}du+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
29

Multiply $-\frac{51}{50}$ times $\frac{1}{2}$

$-\frac{51}{100}\int\frac{1}{u}du+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
30

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

$-\frac{51}{100}\ln\left|u\right|+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
31

Substitute $u$ back for it's value, $x^2+1$

$-\frac{51}{100}\ln\left|x^2+1\right|+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|$
32

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

$-\frac{51}{100}\ln\left|x^2+1\right|+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|+C_0$

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