Be Rational—Use Partial Fractions While Integrating

If you’re in some sort of calculus class, you’ve probably seen lots of different integration problems. You might even encounter some that you can’t immediately solve without some sort of manipulation. If you’re trying to integrate a tricky rational function, you can follow these steps. To illustrate the steps, we’ll look at the following example:

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1) Don’t feel overwhelmed! You can start by focusing on the rational function. Our goal is to simplify the function so we can integrate easily. It might be helpful to temporarily forget that the integral exists. And don’t worry—we’ll remember the integral in our last step.

2) Make sure the degree of the numerator is less than the degree of the denominator. In our example, the degree of the numerator is actually greater than the degree of the denominator, so we need to use polynomial long division to help us simplify. After doing so, we have:

image

3) Factor the denominator. We only need to focus on the left half for this step:

image

4) Write out the partial fraction decomposition. Notice how (x-1) occurs twice. That means we have:

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5) Solve unknown values. All we have to do is solve A, B, and C. There are many different ways to do this, but you might want to multiply both sides of our equation by (x+1)(x-1)2. Then we have something a little easier to solve:

image

If you’re stuck trying to find A, B, and C, you might try plugging in random values for x that will eliminate some terms on the right-hand side of the equation. For instance, try x = -1, or x = 1. See what happens!

You will find A = -1, B = 1, and C= 2.

6) Remember the integral. Now you can re-write the original problem. Using all of the information we’ve obtained, we have:

image

Your hard work paid off—you can integrate now!

#integration #integrals #calculus #CAPS #UNM

 


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