Integration as the Reverse of Differentiation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Integration as the Reverse of Differentiation

Mathematics · Calculus & Analysis · ages 17-18
Name ______________________   Date ____________

  1. A general indefinite integral on one interval includes an arbitrary constant c.

    Circle one:   True   False

  2. Differentiating your integral should give back something different from what you started with.

    Circle one:   True   False

  3. A proposed antiderivative of 9x² - 2x is F(x) = ax³ - x² + 5. What value of a makes F'(x) match the integrand for every x?

    Answer: ______________

  4. Using the reverse power rule for xⁿ, with n ≠ -1, what do you divide by?

    • the new power, n + 1
    • the old power, n
    • the coefficient in front of the term
    • nothing, you multiply by n + 1

  5. What is the general indefinite integral of (x + 2)(x - 2)?

    • x³/3 + 4x + c
    • x³/3 - 4x + c
    • x⁴/4 - 4x + c
    • x³/3 - 4 + c

  6. F and G have the same derivative throughout [0, 2]. If F(0) - G(0) = 7, what is F(2) - G(2)?

    • 14, since x has doubled
    • 0, since derivatives agree
    • 7, since the gap is constant
    • 2, since this is the new x

  7. Ravi is asked to integrate x² and writes 2x. Where is the slip?

    • the answer should have been x² again, since the rule leaves powers alone
    • he forgot only the + c, so 2x + c is correct
    • that is differentiating, and integrating gives x³/3 + c
    • he should have divided by the old power, giving x³/2

  8. What is the integral of 12x²?

    • 24x + c
    • 6x³ + c
    • 12x³ + c
    • 4x³ + c

  9. On an interval that excludes x = 0, what is the general indefinite integral of x⁻²?

    • -2x⁻³ + c
    • -x⁻¹ + c
    • x⁻¹ + c
    • x⁻¹/(-1) is undefined

  10. For x > 0, F'(x) = 3x⁻² and F(1) = 2. Write F(x) = -3x⁻¹ + c. What is c?

    Answer: ______________

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Answer key

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

Integration as the Reverse of Differentiation W1-mt_edtk3ArxRk-s1

  1. True · Differentiating destroys any constant, so working backwards cannot recover it. The + c stands for the constant that could have been anything.
  2. False · Integration is the reverse of differentiation, so differentiating the answer has to return the original expression. That is exactly what makes the check work.
  3. 3 · Differentiating gives F'(x) = 3ax² - 2x. Matching 9x² - 2x requires 3a = 9, so a = 3. The constant 5 differentiates to zero.
  4. the new power, n + 1 · Raising the power gives n + 1, and that new power is what you divide by. Dividing by the old power would undo the wrong step.
  5. x³/3 - 4x + c · Expand the product to x² - 4. Its integral is x³/3 - 4x + c.
  6. 7, since the gap is constant · The difference between two antiderivatives on one interval is constant. Its value remains 7.
  7. that is differentiating, and integrating gives x³/3 + c · Ravi differentiated x² to get 2x. Integrating instead raises the power to 3 and divides by 3, giving x³/3 + c.
  8. 4x³ + c · The power goes from 2 to 3, and 12 is divided by the new power 3, giving 4x³ + c.
  9. -x⁻¹ + c · Raise the power from -2 to -1, then divide by the new power -1. That gives x⁻¹ divided by -1, which is -x⁻¹, or -1/x.
  10. 5 · At x = 1, F(1) = -3 + c = 2, so c = 5.
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