1. Section 8.5 – Partial Fractions

2. White Board Challenge Find a common denominator:

3. White Board Challenge Use the final form to start: Since: Write and solve a System of Equations: Write the final answer:

4. Integration of Rational Fractions We can integrate some rational functions by expressing it as a sum of simpler fractions, called partial fractions, that we already know how to integrate. (We can assume that the degree of the numerator is smaller than the degree of the denominator. Otherwise we could divide and work with the remainder.) From Algebra we can find a common denominator to show: Thus, if we reverse the procedure, we see how to integrate the function on the right side of the equation: Use this method if no previous technique works and the denominator factors nicely.

5. Example 1 Factor the denominator first: Use Partial Fractions to rewrite the integral: Write and solve a System of Equations: Integrate: Since:

6. Example 2 Factor the denominator first: Use Partial Fractions to rewrite the integral: Write and solve a System of Equations: Trick: Multiple each numerator by the other denominators and set it equal to the original numerator: Distribute and combine like terms

7. Example 2 (continued) Factor the denominator first: Use Partial Fractions to rewrite the integral: Integrate:Since:

8. Example 2 EASIER Factor the denominator first: Use Partial Fractions to rewrite the integral: Trick: Ignore the Discontinuity. The “A” fraction is undefined at x=0. Substitute x=0 into the original factored integral to find A. The “B” fraction is undefined at x=1/2. Substitute x=1/2 into the original factored integral to find B. The “C” fraction is undefined at x=-2. Substitute x=-2 into the original factored integral to find C. IGNORE Now find the antiderivative.

9. Example 3 Substitute into the Logistic Model: Separate the variables : Use Partial Fractions: Ignore the Discontinuity: -- ----- Find the antiderivative:Use the initial condition: Solve for P: Most Logistic Model AP exam questions can be answered without solving the differential equation analytically. But seeing it can’t hurt.

10. White Board Challenge Evaluate: