Partial Fractions - Calculus 2

Rational functions are one of the most difficult types of functions to integrate. This is due to the variety of rational functions that exist and the fact that not all of them can be integrated in the same way. In the past we have been able to use a whole arsenal of techniques including u-substitution, the log rule of integration, integration rules for inverse trig functions, integration rules for inverse hyperbolic functions, the newly introduced trig substitution, and more!

But what if none of these integration techniques work? What if we have a rational function composed of a complex polynomial in both the numerator and denominator? (or even just in the denominator?) Well, in most of these cases, a new method known as partial fractions is going to be your best friend.

The method of integration by partial fractions is best used when you want to integrate a rational function that is composed of a complicated polynomial that is at least found in the denominator, but also can be found in the numerator as well. In fact, as long as the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, we will be able to "decompose," or break up that rational function into simple rational functions we call "partial fractions." (If that requirement is not met, then things get a bit messy. We would need to use long division for polynomials from algebra first, and then use the partial fractions method.)

Determining how to break up these rational functions into partial fractions and then integrate them is the heart of this lesson in Calculus 2. The good news is that there isn't really any new "calculus" to learn here. The partial fractions method is really all rooted in skills from algebra. Of course we still need to integrate the resulting partial fractions, but we already know all the integration rules we will need to use to do just that. The bad news on the other hand, is that partial fractions can be a lengthy process depending on the rational function. But let's not waste anymore time and dive into partial fractions! 

In this lesson, you will learn:

• How to identity different types of factors in the denominator of a rational function
• How to decompose, or "break up" a rational function into partial fractions based on the identified factors
• How to integrate the partial fractions that result from the decomposition process

I hope you find this video to be helpful!

-Josh