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Section 8.5 – Partial Fractions

Section 8.5 – Partial Fractions. White Board Challenge. Combine the two fractions into a single fraction: . Find a common denominator:. White Board Challenge. Reverse the process and write as the sum of two fractions in the form: . Use the final form to start:. Since:.

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Section 8.5 – Partial Fractions

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  1. Section 8.5 – Partial Fractions

  2. White Board Challenge Combine the two fractions into a single fraction: Find a common denominator:

  3. White Board Challenge Reverse the process and write as the sum of two fractions in the form: . 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 Use Partial Fractions to rewrite the integral: Evaluate Factor the denominator first: Since: Write and solve a System of Equations: Integrate:

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

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

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

  9. Most Logistic Model AP exam questions can be answered without solving the differential equation analytically. But seeing it can’t hurt. Example 3 A deer population grows logistically with growth constant in a forest with a carrying capacity of 1000 deer. If the initial population is 100 deer, find the particular solution that models the population. Substitute into the Logistic Model: Find the antiderivative: Use the initial condition: Separate the variables : Solve for P: Use Partial Fractions: -- Ignore the Discontinuity: -----

  10. White Board Challenge Evaluate:

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