Section 8.4a. A flashback to Section 6.5… We evaluated the following integral: This expansion technique is the method of partial fractions. Any rational.

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Presentation transcript:

Section 8.4a

A flashback to Section 6.5… We evaluated the following integral: This expansion technique is the method of partial fractions. Any rational function can be written as a sum of basic fractions, called partial fractions, using this method.

Method of Partial Fractions (f(x)/g(x) Proper) Notes: The degree of f(x) must be less than the degree of g(x) (if it is not, do long division!). Also, g(x) must be factorable. 1. Let be a linear factor of. Suppose is the highest power of that divides. Then, to this factor, assign the sum of the partial fractions: Do this for each distinct linear factor of.

Method of Partial Fractions (f(x)/g(x) Proper) Notes: The degree of f(x) must be less than the degree of g(x) (if it is not, do long division!). Also, g(x) must be factorable. 2. Let be a quadratic factor of. Suppose is the highest power of this factor that divides. Then, to this factor, assign the sum of the partial fractions: Do this for each distinct quadratic factor of that cannot be factored into linear factors with real coefficients.

Method of Partial Fractions (f(x)/g(x) Proper) Notes: The degree of f(x) must be less than the degree of g(x) (if it is not, do long division!). Also, g(x) must be factorable. 3. Set the original fraction equal to the sum of all these partial fractions. Clear the resulting equation of fractions and arrange the terms in decreasing powers of. 4. Equate the coefficients of corresponding powers of and solve the resulting equations for the undetermined coefficients. The new fractions should be easier to integrate!!!

Guided Practice Use partial fractions to evaluate the integral. Solve the system!!!

Guided Practice Use partial fractions to evaluate the integral.

Guided Practice Use partial fractions to evaluate the integral. Solve the system!!!

Guided Practice Use partial fractions to evaluate the integral.

Guided Practice Use partial fractions to evaluate the integral. The degree of the numerator is larger than the degree of the denominator… we need long division first:

Guided Practice Use partial fractions to evaluate the integral.

Guided Practice Use partial fractions to evaluate the integral.

Guided Practice Use partial fractions to evaluate the integral. Solve the system!!!

Guided Practice Use partial fractions to evaluate the integral.

Guided Practice Solve the given initial value problem.

Guided Practice Solve the given initial value problem.

Guided Practice Solve the given initial value problem. Initial Condition: Solution: