How to Integrate Using Polynomial Long Division with Examples
When integrating a rational function where the degree of the numerator is greater than or equal to the degree of the denominator, polynomial long division becomes a valuable tool for simplifying the integral. In this guide, we’ll look at how to integrate using polynomial long division and work through a step-by-step example to solidify your understanding.
This article is part of a three-part series on integrals of rational functions. The following infographic illustrates this topic.
Polynomial Long Division
Before integrating, let’s review polynomial long division. This process allows us to rewrite an improper rational function as a sum of a polynomial and a proper fraction, making integration possible with other methods like partial fraction decomposition or completing the square, and maybe even simpler methods.
A proper rational function has a numerator with a degree less than the degree of the denominator. In comparison, an improper rational function has a numerator with a degree greater than or equal to the degree of the denominator. When faced with the latter, polynomial long division helps us write the rational function as the sum of a polynomial and a proper rational function.
Steps to Integrating Using Polynomial Long Division
Step 1: Check That the Rational Function is Improper
Check to make sure the degree of the numerator is greater than or equal to the degree of the denominator. If not, try a different method.
Step 2: Use Polynomial Long Division to Rewrite the Rational Function
We demonstrate this process by rewriting \( \frac{x^2 + 7x + 13}{x^2 + 7x + 12} \).
We first perform polynomial long division:
We now rewrite our function according to the word equation
$$\frac{dividend}{divisor} = quotient + \frac{remainder}{divisor}.$$
In our case, this gives
$$\frac{x^2 + 7x + 13}{x^2 + 7x + 12} = 1 + \frac{1}{x^2 + 7x + 12}.$$
Step 3: Integrate Term-by-Term
The polynomial function can be integrated using the methods covered in the article Basic Integration Problems for Beginners. These methods may also be used to integrate the rational function, but in all likelihood, you will need the techniques covered in The Ultimate Step-by-Step Guide to Solving Integrals Using Substitution or The Ultimate Guide on How to Solve Integrals by Completing the Square to solve it.
Partial fraction decomposition may also be needed. To learn how to find the partial fraction decomposition of a rational function, please check out A Comprehensive Beginner’s Guide to Partial Fraction Decomposition. The article Partial Fraction Decomposition Integration Problems with Solutions: A Complete Tutorial shows how partial fraction decomposition is used to integrate rational functions.
Worked Out Example
We now give an example.
Example 1: Evaluate:
(a) \( \int \frac{2x^3 + 13x^2 + 26x + 11}{x^2 + 6x + 10}dx \)
(b) \( \int \frac{x^2 + 7x + 13}{x^2 + 7x + 12}dx \)
Solution: (a) \( \int \frac{2x^3 + 13x^2 + 26x + 11}{x^2 + 6x + 10}dx \)
We first perform polynomial long division: as follows:
Rewriting the fraction then gives
$$\int 2x + 1 + \frac{1}{x^2 + 6x + 10}dx.$$
By linearity, this is
$$\int 2x + 1 dx + \int \frac{1}{x^2 + 6x + 10}dx.$$
We now complete the square in the denominator of the second integral. Begin by writing 10 as 9 + 1 to get
$$\int 2x + 1 dx + \int \frac{1}{x^2 + 6x + 9 + 1}dx. $$
This is equivalent to
$$\int 2x + 1 dx + \int \frac{1}{(x + 3)^2 + 1}dx. $$
Let \( u = x + 3 \), the \( du = dx \). Substituting into the second integral, we get
$$\int 2x + 1 dx + \int \frac{1}{u^2 + 1}du.$$
Integrating gives
$$x^2 + x + \arctan{u} + C.$$
Back substituting we arrive at a final answer of
$$x^2 + x + \arctan{(x + 3)} + C.$$
(b) \( \int \frac{x^2 + 7x + 13}{x^2 + 7x + 12}dx \)
We first perform polynomial long division: as follows:
Rewriting the fraction then gives
$$\int 1 + \frac{1}{x^2 + 7x + 12}dx.$$
Factoring the denominator, we obtain
$$\int 1 + \frac{1}{(x + 3)(x + 4)}dx.$$
We now need to find the partial fraction decomposition of \( \frac{1}{(x + 3)(x + 4)} \). Setting up the partial fraction decomposition, we get
$$\frac{1}{(x + 3)(x + 4)} = \frac{A}{x + 3} + \frac{B}{x + 4}.$$
Next, multiplying both sides by \( (x + 3)(x + 4) \) gives
$$1 = A(x + 4) + B(x + 3).$$
Now we solve for the coefficients by choosing convenient values of x:
$$x = -3 \Rightarrow 1 = A$$
$$x = -4 \Rightarrow 1 = -B \Rightarrow -1 = B.$$
Thus,
$$\frac{1}{(x + 3)(x + 4)} = \frac{1}{x + 3} – \frac{1}{x + 4}.$$
Our integral is then
$$\int 1 + \frac{1}{x + 3} – \frac{1}{x + 4}dx.$$
Integrating gives
$$x + \ln{|x + 3|} – \ln{|x + 4|} + C.$$
Using logarithmic properties we arrive at a final answer of
$$x + \ln{|\frac{x + 3}{x + 4}|} + C.$$
Conclusion
Polynomial long division is an important tool in integration, especially when dealing with improper rational functions. This guide taught you how to integrate using polynomial long division. By mastering this technique, along with completing the square and partial fraction decomposition, you can, at least in theory, integrate any rational function.
Further Reading
A Comprehensive Beginner’s Guide to Partial Fraction Decomposition – In order to integrate most rational functions you must learn how to perform partial fraction decomposition.
How to Solve a System of Equations Using Gaussian Elimination: A Step-by-Step Guide – The more difficult partial fraction decomposition problems lead to systems that must be solved via Gaussian Elimination.
