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

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Solution

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Final answer to the problem

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

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1

Combining like terms $-x^2$ and $8x^2$

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

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

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

Find the values for the unknown coefficients: $A, B, C, D$. The first step is to multiply both sides of the equation from the previous step by $\left(x^2+1\right)\left(x-3\right)^2$

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

Multiplying polynomials

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

Simplifying

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

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

$\begin{matrix}18=-16A+16B+2C-8D&\:\:\:\:\:\:\:(x=-1) \\ 0=4A+4B+2C-4D&\:\:\:\:\:\:\:(x=1) \\ 38=10C&\:\:\:\:\:\:\:(x=3) \\ 92=-108A+36B+10C-60D&\:\:\:\:\:\:\:(x=-3)\end{matrix}$
7

Proceed to solve the system of linear equations

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

Rewrite as a coefficient matrix

$\left(\begin{matrix}-16 & 16 & 2 & -8 & 18 \\ 4 & 4 & 2 & -4 & 0 \\ 0 & 0 & 10 & 0 & 38 \\ -108 & 36 & 10 & -60 & 92\end{matrix}\right)$
9

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 0 & 0 & -\frac{51}{50} \\ 0 & 1 & 0 & 0 & \frac{7}{50} \\ 0 & 0 & 1 & 0 & \frac{19}{5} \\ 0 & 0 & 0 & 1 & \frac{51}{50}\end{matrix}\right)$
10

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

$\int\left(\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}+\frac{19}{5\left(x-3\right)^2}+\frac{51}{50\left(x-3\right)}\right)dx$
11

Expand the integral $\int\left(\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}+\frac{19}{5\left(x-3\right)^2}+\frac{51}{50\left(x-3\right)}\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately

$\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx+\int\frac{19}{5\left(x-3\right)^2}dx+\int\frac{51}{50\left(x-3\right)}dx$
12

We can solve the integral $\int\frac{19}{5\left(x-3\right)^2}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x-3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x-3$
13

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=dx$
14

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

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

The integral $\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx$ results in: $-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\arctan\left(x\right)$

$-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\arctan\left(x\right)$
16

The integral $\frac{1}{5}\int\frac{19}{u^2}du$ results in: $\frac{-19}{5\left(x-3\right)}$

$\frac{-19}{5\left(x-3\right)}$
17

Gather the results of all integrals

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

Rewrite the fraction $\frac{x}{x^2+1}$ inside the integral as the product of two functions: $x\frac{1}{x^2+1}$

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

We can solve the integral $\int x\frac{1}{x^2+1}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
20

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
21

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\frac{1}{x^2+1}dx}\\ \displaystyle{\int dv=\int \frac{1}{x^2+1}dx}\end{matrix}$
22

Solve the integral

$v=\int\frac{1}{x^2+1}dx$
23

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

$\frac{1}{\sqrt{1}}\arctan\left(\frac{x}{\sqrt{1}}\right)$
24

Now replace the values of $u$, $du$ and $v$ in the last formula

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

We can solve the integral $\int\frac{51}{50\left(x-3\right)}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x-3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x-3$
26

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=dx$
27

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

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

The integral $\frac{1}{50}\int\frac{51}{u}du$ results in: $\frac{51}{50}\ln\left(x-3\right)$

$\frac{51}{50}\ln\left(x-3\right)$
29

Gather the results of all integrals

$\frac{7}{50}\arctan\left(x\right)-\frac{51}{50}\left(x\arctan\left(x\right)-\int\arctan\left(x\right)dx\right)+\frac{-19}{5\left(x-3\right)}+\frac{51}{50}\ln\left(x-3\right)$
30

Solve the product $-\frac{51}{50}\left(x\arctan\left(x\right)-\int\arctan\left(x\right)dx\right)$

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

The integral $\frac{51}{50}\int\arctan\left(x\right)dx$ results in: $\frac{51}{50}x\arctan\left(x\right)-\frac{51}{100}\ln\left(1+x^2\right)$

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

Gather the results of all integrals

$\frac{7}{50}\arctan\left(x\right)-\frac{51}{50}x\arctan\left(x\right)-\frac{51}{100}\ln\left(1+x^2\right)+\frac{51}{50}x\arctan\left(x\right)+\frac{-19}{5\left(x-3\right)}+\frac{51}{50}\ln\left(x-3\right)$
33

Cancel like terms $-\frac{51}{50}x\arctan\left(x\right)$ and $\frac{51}{50}x\arctan\left(x\right)$

$\frac{7}{50}\arctan\left(x\right)-\frac{51}{100}\ln\left(1+x^2\right)+\frac{-19}{5\left(x-3\right)}+\frac{51}{50}\ln\left(x-3\right)$
34

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

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

Final answer to the problem

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

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((-1x^2+8x^2)/(x^2+1)(x-3)^2)dx using partial fractionsSolve integral of ((-1x^2+8x^2)/(x^2+1)(x-3)^2)dx using basic integralsSolve integral of ((-1x^2+8x^2)/(x^2+1)(x-3)^2)dx using u-substitutionSolve integral of ((-1x^2+8x^2)/(x^2+1)(x-3)^2)dx using trigonometric substitution

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

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Main Topic: Integrals by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

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