# Integration of Rational Functions by Partial Fractions

• Jgoshorn1
In summary, the problem involves finding the integral of (x^3+4)/(x^2+4)dx. The first step is to do long division to simplify the expression. This results in x - 4(x-1)/(x^2+4). The next step is to split the integral into two parts and use u-substitution and the Arctan form to solve for each part separately. The 2 in the second part was a typo and should have been 1.
Jgoshorn1

∫(x3+4)/(x2+4)dx

n/a

## The Attempt at a Solution

I know I have to do long division before I can break this one up into partial fractions. So I x3+4 by x2+4 and got x with a remainder of -4x+4 to be written as x+(4-4x/x2+4).

Then I rewrote the initial integral as ∫x + ∫(4-4x)/(x2+4) but then I get stuck from there.
Should the partial fraction be ∫x + ∫(Ax+B)/(x2+4)?
And if so, how would I go about solving that?

##\frac x {x^2+4}## can be done with a u-substitution and ##\frac 2 {x^2+4}##is an Arctan form.

LCKurtz said:
##\frac x {x^2+4}## can be done with a u-substitution and ##\frac 2 {x^2+4}##is an Arctan form.

How/where are you getting x/x2+4 and 2/x2+4 from?

Long division for
$\frac{x^3+4}{x^2+4}$

yields
$x - 4(\frac{x-1}{x^2+4})$.

As you mentioned, you now have
$\int x dx - 4\int \frac{x-1}{x^2+4} dx$.

For the second integral, split up the integrand:
$\int x dx - 4(\int \frac{x}{x^2+4} dx - \int \frac{1}{x^2+4} dx)$.

(This is what LCKurtz did, at least.)

Last edited:
So do you think it was just a typo that he wrote 2/x2+4 instead of 1/x2+4 because that 2 really confused me

Jgoshorn1 said:
So do you think it was just a typo that he wrote 2/x2+4 instead of 1/x2+4 because that 2 really confused me

Yes, the 2 was a typo. At any rate a constant factor wouldn't affect the method of integration.

## What is partial fractions decomposition?

Partial fractions decomposition is a method used to simplify rational functions, which are fractions with polynomials in the numerator and denominator. It involves breaking down the rational function into smaller, simpler fractions.

## When is partial fractions decomposition used?

Partial fractions decomposition is used when integrating rational functions. It can also be used to solve equations involving rational expressions.

## What is the process for integrating rational functions using partial fractions?

The process involves finding the factors of the denominator, setting up a system of equations using the partial fractions, solving for the unknown constants, and then integrating each term separately.

## Why is partial fractions decomposition useful?

Partial fractions decomposition allows us to simplify complex rational functions into smaller, more manageable fractions. This makes it easier to integrate and solve equations involving rational expressions.

## Are there any special cases in partial fractions decomposition?

Yes, there are special cases such as repeated linear factors, irreducible quadratic factors, and repeated irreducible quadratic factors. These cases require additional steps in the decomposition process.

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