Partial Fractions for Rational Integrands · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Split fractions before you integrate

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. Which partial fraction form is correct for 5/(x+2)^2?

    • A/(x+2) + B/(x+2)^2
    • A/(x+2)^2 by itself
    • A/(x-2) + B/(x-2)^2
    • A/(x+2) + B/(x+2)

  2. Before splitting (3x^3 + 2x - 1)/(x^2 - 1) into partial fractions, what should you do first?

    • Factor the denominator and stop there
    • Divide the numerator by the denominator, since the numerator's degree is higher
    • Integrate directly using arctangent
    • Multiply both sides by x^2 - 1 right away

  3. Before splitting (3x^3 + 2x - 1)/(x^2 - 1), what should you do first?

    • Divide, since the top degree is higher
    • Split into partial fractions at once
    • Integrate with arctan directly

  4. For a repeated factor like (x - 2)^3, one term A/(x - 2)^3 is enough.

    Circle one:   True   False

  5. Write 3/(x^2 - x) as A/x + B/(x-1). What is the value of A?

    Answer: ______________

  6. For (x^2 + 1)/(x^2 - 4), the numerator's degree equals the denominator's degree, so you divide first. What is the whole-number quotient you get before dealing with the remainder?

    Answer: ______________

  7. What form should you use to decompose (2x+1)/(x^2(x+3)) into partial fractions?

    • A/x + B/x^2 + C/(x+3)
    • A/x + B/(x+3)
    • (Ax+B)/x^2 + C/(x+3)
    • A/x^2 + B/(x+3)^2

  8. What form splits (2x + 1)/(x^2(x + 3))?

    • A/x + B/(x + 3)
    • A/x + B/x^2 + C/(x + 3)
    • A/x^2 + B/(x + 3)

  9. After splitting into distinct linear pieces, what kind of functions do you integrate?

    • Plain polynomials
    • Powers of x
    • Logarithms like ln|x + 1|

  10. Split 1/(x^2 - x) into -1/x + 1/(x-1), then find the exact value of the integral from x = 2 to x = 3. Give your answer as a decimal, rounded to three decimal places.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Split fractions before you integrate W1-mt_PA8qxs424c-s1

  1. A/(x+2) + B/(x+2)^2 · (x+2)^2 is a repeated linear factor, so you need one term for each power from 1 up to 2, not just the top power alone.
  2. Divide the numerator by the denominator, since the numerator's degree is higher · The numerator has degree 3 and the denominator has degree 2, so the fraction is improper. You divide first to get a polynomial plus a proper remainder, and only then do partial fractions on the remainder.
  3. Divide, since the top degree is higher · The top has degree 3 and the bottom has degree 2, so divide out a polynomial first.
  4. False · You need one term for each power, from 1 up to 3.
  5. -3 · Clearing denominators gives 3 = A(x-1) + Bx. Plugging in x = 0 kills the B term and leaves A by itself.
  6. 1 · Dividing x^2 + 1 by x^2 - 4 gives a quotient of 1 with a remainder of 5, so the fraction becomes 1 + 5/(x^2-4).
  7. A/x + B/x^2 + C/(x+3) · x^2 is a repeated linear factor (x times x), so it needs its own term for power 1 and power 2, and the separate factor (x+3) gets its own single term.
  8. A/x + B/x^2 + C/(x + 3) · The repeated x^2 needs terms for powers 1 and 2, plus one for (x + 3).
  9. Logarithms like ln|x + 1| · Each piece has the form constant over a linear factor, which integrates to a log.
  10. 0.288 · Each piece integrates to a log. Combining the logs and plugging in the bounds gives 2 ln 2 minus ln 3.
Worksheet · LightMySky