Calc 2 Lecture NotesSection 6.4Page 1 of 4

Section 6.4: Integration of Rational Functions Using Partial Fractions

Big idea:In this section, we’ll examine a technique for evaluating integrals that are made of two functions combined through division. Partial fraction decomposition is a technique that allows us to compute the integral of rational functions (i.e., the ratio of two polynomials).

Big skill:. You should be able to decompose various types of rational functions into simpler fractions that can be easily integrated using known formulas.

  1. Warm-up: Adding rational functions.
  2. Combine
  3. Our goal in this section is to figure out how to start with an integrand looking like and convert it to .
  1. Basic example: Linear polynomial in the numerator and a factorizable quadratic polynomial in the denominator: .
  2. A rational function of this form can always be written as: .
  3. Multiply both sides by .
  4. Expand the right-hand side.
  5. Equate like terms to get a system of linear equations in A and B.
  6. Solve the system for A and B.
  7. Shortcut: ; plug in the zeros of each factor to get A and B.
  8. Practice:
  1. Partial Fraction Decomposition for rational functions with distinct linear factors: If a rational function has ndistinct linear factors in the denominator and a numerator polynomial P(x) that is of degree less than n, then that function can be decomposed as:

    for constants c1, c2, …, cn.
  2. Practice:
  1. To check your work using a graphing calculator (with no CAS), or Winplot:
    Graph the integrand and the derivative of the antiderivative on the same screen…
    Y1 = (2X2+1)/(3X3+X2-2X)
    Y2=nDeriv(-0.5ln(X)+0.6ln(X+1)+1.7/3*ln(3X-2),X,X)
    … or use the derive button on the inventory box in winplot…
  1. Caveat: The degree of the numerator must be less than the degree of the denominator for this technique to work  Do long division of the polynomials before integrating if this is not the case.
  2. Practice:
  1. Partial Fraction Decomposition for rational functions with repeated linear factors: If a rational function has a denominator of multiplicity n> 1 and a numerator polynomial P(x) that is of degree less than n, then that function can be decomposed as:

    for constants c1, c2, …, cn.
  2. Practice:

  1. Partial Fraction Decomposition for rational functions with distinct irreducible quadratic factors: If a rational function has ndistinctirreducible quadratic factors in the denominator and a numerator polynomial P(x) that is of degree less than2n, then that function can be decomposed as:

    for constants A1, A2, …, An, B1, B2, …, Bn.,
  2. Practice:
  1. What do you think happens for a rational function with repeated irreducible quadratic factors?