1. Partial Fractions Day 2 Chapter 7.4 April 3, 2007

2. Integrate: Chapter 7.4 April 3, 2007

3. The Arctangent formula (also see day 9 notes) The “new” formula: When to use? If the polynomial in the denominator does not have real roots (b 2 -4ac < 0) then the integral is an arctangent, we complete the square and integrate…. For example:

4. What if the polynomial has real roots? That means we can factor the polynomial and “undo” the addition! To add fractions we find a common denominator and add: we’ll work the other way…. The denominator of our rational function factors into (t - 4)(t +1) So in our original “addition,” the fractions were of the form:

6. Examples:

7. Each of these integrals involved linear factors What if a factor is repeated? For example: The “x” factor is repeated, so in our original addition, we could have had each of the “reduced” fractions: Clearing our denominators, we get:

8. To Solve for A, B, and C, again we choose x carefully:

9. Using this information, our original integral becomes:

10. Example:

11. We may also have expressions with factors of higher powers: We apply the same concept as when there are linear factors, we undo the addition using REDUCED fractions.

12. We have To solve for B and C, we will match coefficients From

13. The Integration:

14. Example: