Integration by Parts · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Trade one integral for an easier one

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

  1. For ∫(2x+1)e^(3x) dx, take u = 2x+1 and dv = e^(3x) dx. Which choice of v has derivative e^(3x)?

    • v = e^(3x)/3
    • v = 3e^(3x)
    • v = e^(3x)
    • v = e^(3x)/9

  2. Which equation is the integration by parts formula?

    • Integral of u dv = u v + integral of v du
    • Integral of u dv = u v - integral of v du
    • Integral of u dv = integral of v du - u v

  3. For x > 0, which split turns ∫x² ln x dx into a product minus a power integral containing no logarithm?

    • u = x², dv = ln x dx
    • u = 1, dv = x² ln x dx
    • u = ln x, dv = x² dx

  4. For ∫x e^x dx, which split lowers the polynomial degree and uses a known exponential antiderivative for v?

    • u = x, dv = e^x dx
    • u = e^x, dv = x dx
    • u = x e^x, dv = dx

  5. Using radians, let I = ∫e^(2x)sin x dx and J = ∫e^(2x)cos x dx. Parts gives I = e^(2x)sin x/2-J/2. Which second relation follows by taking u = cos x in J? Relations are up to constants.

    • J = e^(2x)cos x/2-I/2
    • J = e^(2x)sin x/2+I/2
    • J = e^(2x)cos x/2+I/2
    • J = e^(2x)cos x/2+2I

  6. Using radians, let I = ∫e^x cos x dx. Two parts applications give I = e^x(cos x+sin x)-I, up to a constant. Which step isolates I?

    • Subtract I from both sides, then divide by 2
    • Add I to both sides, then divide by 2
    • Add I to both sides, then divide by -2

  7. Using radians, what is ∫x cos x dx?

    • x sin x + cos x + C
    • x sin x - cos x + C
    • x cos x + sin x + C

  8. Let f have a continuous derivative on [0,2], with f(0) = 3, f(2) = 4 and ∫ from 0 to 2 of f(x) dx = 5. Find ∫ from 0 to 2 of (2x+1)f'(x) dx.

    • 13
    • 7
    • 17
    • -7

  9. Using the polynomial as u and dv = e^x dx each round, how many parts applications reduce ∫x²e^x dx to a plain exponential integral?

    • 1
    • 2
    • 3

  10. For x > 0, what is ∫(ln x)² dx after repeated integration by parts?

    • x(ln x)² + 2x ln x + 2x + C
    • x(ln x)² - 2x ln x - 2x + C
    • x(ln x)² - 3x ln x + 2x + C
    • x(ln x)² - 2x ln x + 2x + C
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Answer key

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

Trade one integral for an easier one W1-mt_v6jZVs2_tf-s1

  1. v = e^(3x)/3 · Differentiating e^(3x) introduces a factor 3. Dividing by 3 cancels that factor.
  2. Integral of u dv = u v - integral of v du · Integrating the product rule and subtracting the integral of v du gives the formula.
  3. u = ln x, dv = x² dx · With u = ln x and v = x³/3, the product v du is x² dx/3, which contains no logarithm.
  4. u = x, dv = e^x dx · Taking u = x gives du = dx. With v = e^x, the remaining integral is ∫e^x dx.
  5. J = e^(2x)cos x/2+I/2 · For J, du = -sin x dx and v = e^(2x)/2. Subtracting the remaining integral therefore adds I/2.
  6. Add I to both sides, then divide by 2 · Adding I gives 2I = e^x(cos x+sin x). Divide by 2, then include the arbitrary constant.
  7. x sin x + cos x + C · Parts gives x sin x-∫sin x dx. The antiderivative of sin x is -cos x, so subtracting it gives +cos x.
  8. 7 · The boundary product is 5(4)-1(3) = 17. Subtract 2 times the supplied integral: 17-10 = 7.
  9. 2 · The polynomial factor becomes 2x after one application and 2 after the next. Integrate the remaining exponential directly.
  10. x(ln x)² - 2x ln x + 2x + C · Parts gives x(ln x)²-2∫ln x dx. Substitute x ln x-x and distribute -2 to obtain x(ln x)²-2x ln x+2x+C.
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