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:
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:
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.
- ,
- .
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.
- ,
- ,
- .
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
henceDenominators 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.
- ,
- ,
- .
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 findThe 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.
- ,
- ,
- ,
- ,
- .
From this we can now rewrite our integral as Each term of this new integrand is easy to evaluate, write
SoThe 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.