Integrating Partial Fractions

➗ Long Division

Either when the numerator is greater than or equal to the denominator in power

Ex: ∫ (x² - 8 / x + 3) dx

  1. Long division

  2. Don’t forget about the reminder / divider

  3. Integrate with result

Ex: ∫ (x² / x² - 3x + 2) dx

  • Even if the denominator looks bigger, it still is the divider

🧩 Decomposition

When the denominator is greater than the numerator in power

Ex: ∫ (5x - 1 / x² - 2x - 8) dx

  • Hint: Easily can factor the denominator

  1. Factor the denominator

    Ex: (x+2) (x-4)

  2. Write one partial fraction for each of those factors

  3. Multiply through by the bottom so we no longer have fractions

  4. Now find the constants A and B

    Ex: A = 11/6, B = 19/6

  5. Integral the new partial fractions

    Ex: ∫ (11 / 6(x+2)) dx ∫ (19 / 6(x-4)) dx

  6. Add C

    Ex: 11/6 ln |x + 2| + 19/6 ln |x - 4| + C

⏹ Completing the Square

When there is a 1 or dx in the numerator

Ex: ∫ (1 / x² + 6x + 14) dx

  • Hint: Can’t easily factor the denominator

  1. Identify b and c

    Ex: b = 6, c = 14

  2. Plug into formula

    (x + b/2)² - (b/2)² + C

    Ex: ( x + 6/2)² - (6/2)² + 14 → simplify

  3. Plug in new equation in the denominator

    Ex: ∫ (1 / (x + 3)² + 5) dx

  4. Find familiarity in a specific arctrig

    1. Arctan → ∫ (du / a² + u²) = 1/a arctan(u/a) + C

    2. Arcsin → ∫ (du / √a² + u²) = arcsin(u/a) + C

  5. Plug into arctrig

    Ex: a² = (x + 3)² SO a = (x + 3)

    Ex: u² = 5 SO u = √5

  6. Add C

    Ex: 1/√5 arctan (x + 3/√5) + C