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Math10B Lecture 19 Notes

Dec 8, 2025

Overview

  • Topic: Using Partial Fractions to Evaluate Integrals.
  • Covers examples illustrating partial fraction decomposition, factoring, solving for coefficients, integrating resulting terms.
  • Emphasizes reduced form, proper vs. improper rational functions, and handling repeated or irreducible quadratic factors.

Example A: Integrate (x - 4) / (x^2 - 5x + 6) dx

  • Factor denominator: x^2 - 5x + 6 = (x - 3)(x - 2).
  • Distinct linear factors → decomposition: A/(x - 3) + B/(x - 2).
  • Combine to get numerator identity: x - 4 = A(x - 2) + B(x - 3).
  • Equate coefficients → system:
    • For x: 1 = A + B
    • Constant: -4 = -2A - 3B
  • Solve: A = -1, B = 2.
  • Integral result:
    • ∫(x - 4)/(x^2 - 5x + 6) dx = -ln|x - 3| + 2 ln|x - 2| + C.

Example B: Integrate (x^2 - 2x + 1) / (x^4 - x^2) dx

  • Factor denominator: x^4 - x^2 = x^2(x^2 - 1) = x^2(x - 1)(x + 1).
  • Factor numerator: x^2 - 2x + 1 = (x - 1)^2.
  • Cancel common (x - 1) factor → reduced integrand: (x - 1)/(x^2(x + 1)).
  • Denominator factors: x, x (repeated), x + 1 → three partial terms.
  • Decomposition form: A/x + B/x^2 + C/(x + 1).
  • Combine and equate numerators: x - 1 = A x(x + 1) + B(x + 1) + C x^2.
  • Collect coefficients → system:
    • x^2: 0 = A + C
    • x^1: 1 = A + B
    • constant: -1 = B
  • Solve: B = -1, A = 2, C = -2.
  • Decomposed integrand: 2/x - 1/x^2 - 2/(x + 1).
  • Integrate termwise:
    • ∫2/x dx = 2 ln|x|
    • ∫(-1/x^2) dx = +1/x
    • ∫(-2/(x + 1)) dx = -2 ln|x + 1|
  • Final answer: 2 ln|x| + 1/x - 2 ln|x + 1| + C.

Remark About Alternative Grouping

  • Treating a repeated linear factor x^2 as a single quadratic with numerator ax + b is algebraically equivalent.
  • Splitting ax/(x^2) + b/(x^2) yields the earlier A/x + B/x^2 form.
  • Prefer the split form because it produces integrable simple terms directly.

Example C: Integrate (x^2 + x) / ((x^2 + 1)(x^2 + 4)) dx

  • Denominator factors are irreducible quadratics: x^2 + 1 and x^2 + 4.
  • Decomposition form: (Ax + B)/(x^2 + 1) + (Cx + D)/(x^2 + 4).
  • Combine and equate numerators: x^2 + x = (Ax + B)(x^2 + 4) + (Cx + D)(x^2 + 1).
  • Expand and collect coefficients for x^3, x^2, x, constant.
  • Equations from coefficients:
    • x^3: 0 = A + C
    • x^2: 1 = B + D
    • x^1: 1 = 4A + C
    • constant: 0 = 4B + D
  • Solve system:
    • From A + C = 0 and 1 = 4A + C → A = 1/3, C = -1/3.
    • From 1 = B + D and 0 = 4B + D → B = 1/3, D = 4/3.
  • Partial fractions:
    • (1/3 x + 1/3)/(x^2 + 1) + (-1/3 x + 4/3)/(x^2 + 4).
  • Split each numerator into two simpler fractions:
    • 1/3 * x/(x^2 + 1) + 1/3 * 1/(x^2 + 1)
    • -1/3 * x/(x^2 + 4) + 4/3 * 1/(x^2 + 4)
  • Integrate each term:
    • ∫x/(x^2 + a) dx → (1/2) ln(x^2 + a) via u = x^2 + a.
    • ∫1/(x^2 + a) dx → (1/√a) arctan(x/√a).
  • Final antiderivative:
    • (1/6) ln(x^2 + 1) - (1/3) arctan(x)
      • (1/6) ln(x^2 + 4) + (2/3) arctan(x/2) + C
    • (Coefficients combined from decomposition results.)

Example D: Integrate x^2 / (x + 1) dx (Improper Rational Function)

  • Top degree (2) > bottom degree (1) → improper rational function.
  • First perform polynomial division: x^2 ÷ (x + 1).
  • Division result: x - 1 with remainder 1 → x^2/(x + 1) = x - 1 + 1/(x + 1).
  • Integrate termwise:
    • ∫(x - 1) dx = x^2/2 - x
    • ∫1/(x + 1) dx = ln|x + 1|
  • Final answer: x^2/2 - x + ln|x + 1| + C.

Key Terms and Definitions

  • Rational Function: polynomial divided by polynomial.
  • Proper Rational Function: numerator degree < denominator degree.
  • Improper Rational Function: numerator degree ≥ denominator degree; perform polynomial division first.
  • Reduced Form: numerator and denominator share no common polynomial factor.
  • Partial Fraction Decomposition: express rational function as sum of simpler fractions based on factor types:
    • Distinct linear factors: constants over (x - r).
    • Repeated linear factors: terms A1/(x - r) + A2/(x - r)^2 + ...
    • Irreducible quadratic factors: (Ax + B)/(quadratic).
  • Integration techniques used:
    • Logarithm rule: ∫1/(u) du = ln|u|.
    • Substitution: for u = x^2 + a when integrating x/(x^2 + a).
    • Arctangent rule: ∫1/(x^2 + a^2) dx = (1/a) arctan(x/a).
    • Power rule: integrate polynomials termwise.

Action Items / Next Steps (If Studying)

  • Practice decomposing rational functions: distinct linear, repeated linear, irreducible quadratics.
  • Solve coefficient systems by equating coefficients and by plugging convenient x-values.
  • Review substitution and arctangent integrals for 1/(x^2 + a).
  • Practice polynomial division and algebraic manipulations to convert improper rational functions to proper form.

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