Overview
This lecture covers partial fraction decomposition as a method to integrate rational functions, including all main cases (distinct linear, repeated linear, irreducible quadratic, repeated quadratic). It demonstrates practice through examples and discusses both algebraic setup and calculus steps.
Partial Fractions: Introduction & Theory
- Partial fractions decompose a rational function (polynomial over polynomial) into simpler fractions for easier integration.
- Every polynomial can be factored into linear and/or irreducible quadratic factors.
- Any rational function where the degree of the numerator is less than the denominator can be written as partial fractions.
- If the numerator's degree is equal to or greater than the denominator's, perform long division first.
Distinct Linear Factors (Case 1)
- If the denominator has distinct linear factors, express as a sum: ( \frac{A}{factor_1} + \frac{B}{factor_2} + ... )
- To set up, factor the denominator completely; assign a constant numerator to each linear factor.
- Find constants by equating numerators after clearing denominators ("cover-up" method or plugging in suitable x-values).
Repeated Linear Factors (Case 2)
- For repeated linear factors, include terms up to the highest power: ( \frac{A}{(x-a)} + \frac{B}{(x-a)^2} + ... )
- Proceed as above, assigning a separate constant numerator for each exponent.
Irreducible Quadratic Factors (Case 3)
- For unfactorable quadratics, use numerators of one degree lower: ( \frac{Bx+C}{ax^2+bx+c} )
- Combine with other types if present in denominator.
Repeated Irreducible Quadratics (Case 4)
- For repeated irreducible quadratics, include fractions up to the repeated power: ( \frac{Bx+C}{(ax^2+bx+c)} + \frac{Dx+E}{(ax^2+bx+c)^2} + ... )
Process & Example Steps
- Always factor denominator fully into linear/irreducible quadratic terms.
- Set up a sum of fractions corresponding to denominator factors as above.
- Obtain common denominator, set numerators equal, and solve for constants (via system of equations or plugging in strategic x-values).
- After decomposition, integrate each term (logarithms for linear, arctangent for quadratic, powers as needed).
- For repeated factors or higher-degree quadratics, you may need substitution or trigonometric substitution.
Special Techniques
- The "cover-up method" helps rapidly find coefficients for linear factors.
- Use substitution for linear factors with coefficients, or for integrals involving terms like ( \frac{1}{x^2+a^2} ) (gives arctangent).
- Trigonometric substitution may be required for integrals with irreducible quadratics.
Key Terms & Definitions
- Polynomial — An expression with non-negative integer powers of x and real coefficients.
- Rational function — A ratio of two polynomials ( \frac{R(x)}{Q(x)} ).
- Irreducible quadratic — A quadratic that cannot be factored over the real numbers.
- Partial fraction decomposition — Expressing a rational function as a sum of simpler fractions.
Action Items / Next Steps
- Practice factoring polynomials and identifying factor types.
- Work through assigned homework examples of each case.
- Review integration techniques for logarithms, arctangent, and trigonometric substitution.
- Complete any exercises or readings assigned at the end of Section 7.4.