We discuss an approach that allows us to integrate rational functions.

### Basics of polynomial and rational functions

In this course, we are attempting to learn to work with as many functions as possible.
A basic class of functions are *polynomial functions*:

**polynomial function**in the variable is a function which can be written in the form where the ’s are all constants (called the

**coefficients**) and is a whole number (called the

**degree**when ). The domain of a polynomial function is .

As we will see, there is a fact about polynomials that is of critical importance for this section:

**Polynomials are equal as functions if and only if their
coefficients are equal.**

In the world of mathematics, polynomials are a generalization of “integers,” and rational numbers are fractions of integers. This brings us to our next definition:

**rational function**in the variable is a function the form where and are polynomial functions. The domain of a rational function is all real numbers except for where the denominator is equal to zero.

### Denominators with distinct linear factors

We are already skilled at working with polynomials, we can differentiate and
integrate any polynomial function. Being able to integrate *any* rational function is
the next logical step in our (rather ambitious) quest to integrate *all* functions. Let’s
dig right in with an example.

**polynomials are equal as functions if and only if their coefficients are equal**, we may rewrite this as

**two**equations:

Solving these three equations for and we find

- ,
- .

From this we can now rewrite our integral as

What we have seen is part of a general technique of integration called “partial fractions” that, in principle, allows us to integrate any rational function.

#### The general technique for distinct linear factors

Suppose you wish to compute where and are both polynomial functions, the degree
of is less than the degree of , and factors into **distinct** linear factors:
then we can **always** write The right-hand side of the equation above is
easy to antidifferentiate, as we can integrate it term-by-term and hence

### Denominators with repeated linear factors

Here we work as we did before, except we add an extra variable for each of the repeated factors. Let’s do an example.

**polynomials are equal as functions if and only if their coefficients are equal**, we may rewrite this as

**three**equations:

Solving these three equations for , and we find

- ,
- ,
- .

From this we can now rewrite our integral as

#### The general technique for repeated linear factors

Suppose you wish to compute where and are both polynomial functions, the
degree of is less than the degree of , and factors into **repeated** linear
factors: then we can **always** write The right-hand side of the equation
above is easy to antidifferentiate, as we can integrate it term-by-term and

hence

### Denominators with distinct irreducible quadratic factors

Here is a fact about polynomials:

Remember, a **root** is where a polynomial is zero. The theorem above is
a deep fact of mathematics. The great mathematician Gauss proved the
theorem in 1799 for his doctoral thesis. This fact can be used to show the
following:

**Every polynomial function will factor as a product of
linear terms and irreducible quadratic terms over the real
numbers.**

So now let’s work an example where the denominator of our rational function has distinct quadratic factors.

**polynomials are equal as functions if and only if their coefficients are equal**, we may rewrite this as

**three**equations:

Solving these three equations for , and we find

- ,
- ,
- .

From this we can now rewrite our integral as The first term of this new integrand is easy to evaluate. We find The second term is not hard, but takes several steps and uses substitution techniques.

The integrand has a quadratic in the denominator and a linear term in the numerator. This leads us to try substitution. Let

However, the numerator is , not ! We can bypass this difficulty by adding “” in the form of “.”

We can now integrate the first term with substitution, leading to The final term can be integrated using arctangent. First, complete the square in the denominator: then use a substitution of to find Let’s start at the beginning and put all of the steps together.

breaking this integral up we find

and antidifferentiating we find

#### The general technique for distinct quadratic factors

Suppose you wish to compute where and are both polynomial functions, the degree
of is less than the degree of , and factors into **distinct** irreducible quadratic factors:
then we can **always** write The right-hand side of the equation can be
antidifferentiated, though it is not always “easy.”

### Denominators with repeated quadratic factors

For completeness sake, we will work a problem with repeated quadratic factors.

**polynomials are equal as functions if and only if their coefficients are equal**, we may rewrite this as

**five**equations:

Solving these three equations for , , , , and we find

- ,
- ,
- ,
- ,
- .

From this we can now rewrite our integral as Each term of this new integrand is easy to evaluate, write

So

#### The general technique for repeated quadratic factors

Suppose you wish to compute where and are both polynomial functions, the degree
of is less than the degree of , and factors into **distinct** irreducible quadratic factors:
then we can **always** write The right-hand side of the equation can be
antidifferentiated, though it is not always “easy.”

### Reducing rational functions

When computing all of the techniques above rely on the fact that the degree of is less than the degree of . What if this is not the case? Use long-division.

As with many other problems in calculus, it is important to remember that one is not expected to “see” the final answer immediately after seeing the problem. Rather, given the initial problem, we break it down into smaller problems that are easier to solve. The final answer is a combination of the answers of the smaller problems.