# Cilian's question at Yahoo Answers regarding integration by partial fractions

• MHB
• Chris L T521
In summary, the integral is evaluated using partial fractions, and the first two integrals are straightforward; the last integral requires a substitution.
Chris L T521
Gold Member
MHB
Here is the question:

Cilian said:
Find the integral
$\int\frac{19x^2-x+4}{x(1+4x^2)}$

Here is a link to the question:

Integral of ((19x^2)-x+4)/(x(1+4(x^2)))? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

Hello Cilian,

We're going to use partial fractions to evaluate this integral. With that said, we note that the partial fraction decomposition will take on the form

$\frac{19x^2-x+4}{x(1+4x^2)} = \frac{A}{x}+\frac{Bx+C}{1+4x^2}$

Multiplying both sides by the common denominator yields

$19x^2-x+4 = A(1+4x^2) + (Bx+C)x$

We now simplify the right hand side and group like terms to get

$19x^2-x+4 = (4A+B)x^2+Cx+A$

If we now compare the coefficients of both sides, we have the following system of equations:

\left\{\begin{aligned} 4A + B &= 19 \\ C &= -1 \\ A &= 4 \end{aligned}\right.

Luckily for us, we already have two of the solutions, so it follows now that $4(4)+B = 19 \implies B=3$. Therefore, we now see that

$\frac{19x^2-x+4}{x(1+4x^2)} = \frac{4}{x} + \frac{3x-1}{1+4x^2}$

Hence, we now see that

$\int \frac{19x^2-x+4}{x(1+4x^2)}\,dx = \int\frac{4}{x}\,dx + \int\frac{3x-1}{1+4x^2}\,dx = \color{red}{\int\frac{4}{x}\,dx} + \color{blue}{\int\frac{3x}{1+4x^2}\,dx} - \color{green}{\int\frac{1}{1+4x^2}\,dx}$

The first integral is rather straightforward; you should see that

$\int\frac{4}{x}\,dx = \color{red}{4\ln|x|+C}$

Next, for the second integral, we make a substitution: $u=1+4x^2\implies \,du =8x\,dx \implies \dfrac{du}{8}=x\,dx$. Thus,

$\int\frac{3x}{1+4x^2}\,dx = \frac{3}{8}\int\frac{1}{u}\,du = \frac{3}{8}\ln|u|+C = \color{blue}{\frac{3}{8}\ln(1+4x^2)+C}$

(Note here that we can drop absolute values since $1+4x^2>0$ for any $x$.)

For the last integral, we need to note that

$\int\frac{1}{1+4x^2}\,dx = \int\frac{1}{1+(2x)^2}\,dx$

To integrate, we make the substitution $u=2x\implies \,du = 2\,dx \implies \dfrac{du}{2}=\,dx$. Thus,

$\int\frac{1}{1+4x^2}\,dx = \frac{1}{2}\int\frac{1}{1+u^2}\,du = \frac{1}{2}\arctan(u)+C = \color{green}{\frac{1}{2}\arctan(2x)+C}$

Therefore, putting everything together, we have that

$\int\frac{19x^2-x+4}{x(1+4x^2)}\,dx = 4\ln|x| + \frac{3}{8}\ln(1+4x^2) - \frac{1}{2}\arctan(2x) + C$

I hope this makes sense!

## What is integration by partial fractions?

Integration by partial fractions is a method used in calculus to break down a rational function into simpler fractions that can be integrated separately. This is useful for solving integrals that would otherwise be difficult or impossible to solve.

## How do you find the partial fraction decomposition of a rational function?

The steps for finding the partial fraction decomposition of a rational function are as follows:

1. Factor the denominator of the rational function into its irreducible factors.
2. Write the rational function as a sum of fractions, with each fraction having one of the irreducible factors as its denominator.
3. Determine the coefficients for each fraction by equating the corresponding terms in the original rational function and the decomposed fractions.

Once the partial fraction decomposition is found, each fraction can be integrated separately.

## Can any rational function be integrated using partial fractions?

No. The rational function must have a proper fraction (where the degree of the numerator is less than the degree of the denominator) in order for integration by partial fractions to be applicable. If the rational function is improper, it can be divided into a polynomial and a proper fraction before using the partial fraction method.

## How is integration by partial fractions used in real-world applications?

Integration by partial fractions is used in various fields of science and engineering, such as physics, chemistry, and electrical engineering. It can be used to solve problems involving rates of change, volumes, and areas under curves, among others.

## Are there any alternative methods for solving integrals besides partial fractions?

Yes. Some other methods for solving integrals include substitution, trigonometric substitution, and integration by parts. The choice of method depends on the specific integral being solved and the preferences of the individual.

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