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. A partial-fraction identity represents (x+1)/(x(x-4)). At which real inputs can the original fraction be evaluated?

    • All real x except 0 and 4
    • All real x, including 0 and 4
    • All real x except -1 and 4
    • All real x except 0 and -4

  2. For x≠-1,2, recombine 2/(x+1)+3/(x-2) over (x+1)(x-2). Which numerator results?

    • 5x + 1
    • x - 7
    • 5x - 7
    • 5x - 1

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

    • Divide to obtain a polynomial plus a proper remainder.
    • Split at once without separating out a polynomial quotient.
    • Apply the arctangent formula before changing the fraction.

  4. On an interval avoiding x=-2, an antiderivative of 5/(x+2)^2 is 5 ln|x+2|+C.

    Circle one:   True   False

  5. A split piece is 6/(3x-2). Its antiderivative is k ln|3x-2|+C. What is k?

    Answer: ______________

  6. Divide 2x^3+x^2+5 by x^2+1. Write the remainder as mx+n. What is m?

    Answer: ______________

  7. Which full standard template fits (x+4)/(x(x^2+9)), using a constant numerator for the linear factor and a linear numerator for the quadratic?

    • A/x + (Bx + C)/(x^2 - 9)
    • A/x + (Bx + C)/(x^2 + 9)
    • A/x^2 + (Bx + C)/(x^2 + 9)
    • A/x + (Bx + C)/(x^2 + 9)^2

  8. Which full standard template uses a separate constant numerator over every required linear-factor power for (3x+2)/((x-1)^2(x+5))?

    • A/(x-1) + B/(x-1)^3 + C/(x+5)
    • A/(x-1) + B/(x-1)^2 + C/(x+5)^2
    • A/(x-1) + B/(x-1)^2 + C/(x+5)

  9. A split quadratic piece is (6x+4)/(x^2+4). Which expression is an antiderivative?

    • 3 ln(x^2+4) + 2 arctan(x/2) + C
    • 6 ln(x^2+4) + 4 arctan(x/2) + C
    • 3 ln(x^2+4) + 2 arctan(2x) + C

  10. Evaluate the definite integral of 2/(x(x-1)) from x=2 to x=4. Use 2/(x(x-1))=-2/x+2/(x-1). Give a decimal rounded to three 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. All real x except 0 and 4 · The denominator x(x-4) is zero at x=0 and x=4. Those inputs are excluded. A zero numerator at x=-1 is allowed because the denominator is nonzero there.
  2. 5x - 1 · The common numerator is 2(x-2)+3(x+1)=2x-4+3x+3=5x-1.
  3. Divide to obtain a polynomial plus a proper remainder. · The numerator has degree 3 and the denominator has degree 2. Division separates the polynomial quotient from the proper remainder.
  4. False · The proposed logarithm differentiates to 5/(x+2). The squared denominator instead gives the antiderivative -5/(x+2)+C.
  5. 2 · Differentiating ln|3x-2| gives 3/(3x-2). To obtain numerator 6, use k=6/3=2.
  6. -2 · The quotient is 2x+1. Subtracting (2x+1)(x^2+1) leaves -2x+4, so m=-2.
  7. A/x + (Bx + C)/(x^2 + 9) · The linear factor x takes A/x. Since x^2+9 has no real roots, its numerator must allow Bx+C.
  8. A/(x-1) + B/(x-1)^2 + C/(x+5) · The repeated factor x-1 requires denominator powers 1 and 2. The distinct factor x+5 requires one first-power term. Give each term a separate constant numerator.
  9. 3 ln(x^2+4) + 2 arctan(x/2) + C · The 6x term integrates to 3 ln(x^2+4). With a=2, the constant term integrates to (4/2) arctan(x/2). Add the results and one constant C.
  10. 0.811 · An antiderivative is 2 ln|(x-1)/x|. The upper value minus the lower value is 2 ln(3/2), which rounds to 0.811.
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