![image0.png](https://www.dummies.com/wp-content/uploads/314691.image0.png)
For each quadratic factor in the denominator that’s raised to the third power, add three partial fractions in the following form:
![image1.png](https://www.dummies.com/wp-content/uploads/314692.image1.png)
Generally speaking, when a linear factor is raised to the nth power, add n partial fractions. For example, suppose that you want to integrate the following expression:
![image2.png](https://www.dummies.com/wp-content/uploads/314693.image2.png)
This expression contains all linear factors, but one of these factors (x + 5) is nonrepeating and the other (x – 1) is raised to the third power. Set up your partial fractions this way:
![image3.png](https://www.dummies.com/wp-content/uploads/314694.image3.png)
Which will yield:
![image4.png](https://www.dummies.com/wp-content/uploads/314695.image4.png)
As you can see, this example adds one partial fraction to account for the nonrepeating factor and three to account for the repeating factor.
When you start out with a linear factor, using partial fractions leaves you with an integral in the following form:
![image5.png](https://www.dummies.com/wp-content/uploads/314696.image5.png)
Integrate all these cases by using the variable substitution u = ax + b so that du = a dx and
![image6.png](https://www.dummies.com/wp-content/uploads/314697.image6.png)
This substitution results in the following integral:
![image7.png](https://www.dummies.com/wp-content/uploads/314698.image7.png)
Here are a few examples:
![image8.png](https://www.dummies.com/wp-content/uploads/314699.image8.png)