Integration of Rational Functions

This presentation is about integrating rational functions such as…

First let’s do some easy ones

Find the antiderivative Solution

Find the antiderivative Solution ♠♣ Recall the formula:

Integrate SolutionNotice… top derivative of bottom equals

Find Solution This observation allows us to split the difficult integral into two simple integrals. This leaves just one question: how does one find identities like this? The process of breaking a fraction into simpler fractions is called the method of partial fractions. We make an interesting observation:

The following discussion is about algebra. Not calculus.

Let’s consider the problem of splitting apart the following rational function. Now let’s split it apart somehow! Simplify: These are equal Conclusion:

Let’s review what happened.

Example: Integrate SolutionUsing partial fractions: (Done on previous slide)

Long DivisionThere are situations where partial fractions won’t work. Consider

Performing long division with Set up long division as in elementary arithmetic. Subtract. We can finally write:

Example Integrate: Conclude Now we do partial fractions.