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Quiz: Integral Properties and Techniques

Test your understanding of integral properties and techniques with these review questions.

1. The property ∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx = ∫ₐᶜ f(x) dx is called:

  1. The Sum Rule
  2. The Additivity Property
  3. The Product Rule
  4. The Chain Rule
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The correct answer is B. The Additivity Property allows splitting an integral at any point c between a and b, or combining adjacent integrals. The areas add together.

Concept Tested: Additivity Property

2. What is ∫ₐᵃ f(x) dx?

  1. f(a)
  2. 2f(a)
  3. 0
  4. Undefined
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The correct answer is C. When the limits of integration are equal, the integral is zero. There's no "width" to the region, so no area.

Concept Tested: Zero Width Integral

3. If ∫ₐᵇ f(x) dx = 5, then ∫ᵇᵃ f(x) dx equals:

  1. 5
  2. −5
  3. 0
  4. 10
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The correct answer is B. Reversing the limits of integration negates the integral: ∫ᵇᵃ f(x) dx = −∫ₐᵇ f(x) dx = −5.

Concept Tested: Reversing Limits

4. If f is an even function, then ∫₋ₐᵃ f(x) dx equals:

  1. 0
  2. 2∫₀ᵃ f(x) dx
  3. ∫₀ᵃ f(x) dx
  4. −2∫₀ᵃ f(x) dx
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The correct answer is B. For even functions (symmetric about y-axis), the integral from −a to a equals twice the integral from 0 to a.

Concept Tested: Even Function Integral

5. If f is an odd function, then ∫₋ₐᵃ f(x) dx equals:

  1. 0
  2. 2∫₀ᵃ f(x) dx
  3. ∫₀ᵃ f(x) dx
  4. a·f(0)
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The correct answer is A. For odd functions (symmetric about origin), the positive and negative areas cancel, giving ∫₋ₐᵃ f(x) dx = 0.

Concept Tested: Odd Function Integral

6. To evaluate ∫2x·cos(x²) dx, the best technique is:

  1. Product Rule
  2. u-substitution with u = x²
  3. Integration by parts
  4. Partial fractions
Show Answer

The correct answer is B. Notice that 2x is the derivative of x². Let u = x², du = 2x dx. The integral becomes ∫cos(u) du = sin(u) + C = sin(x²) + C.

Concept Tested: u-Substitution

7. When using u-substitution on a definite integral, you should:

  1. Always back-substitute at the end
  2. Change the limits of integration to u-values
  3. Ignore the limits until the end
  4. Either change limits OR back-substitute, but not both
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The correct answer is D. You can either change the limits to u-values and evaluate directly, OR keep original limits and back-substitute at the end. Both work; don't mix them.

Concept Tested: Changing Bounds

8. The average value of f on [a, b] is given by:

  1. [f(a) + f(b)]/2
  2. (1/(b−a)) ∫ₐᵇ f(x) dx
  3. ∫ₐᵇ f(x) dx
  4. (b−a) ∫ₐᵇ f(x) dx
Show Answer

The correct answer is B. The average value formula is (1/(b−a)) ∫ₐᵇ f(x) dx. It represents the height of a rectangle with the same area as the region under f.

Concept Tested: Average Value Formula

9. What is ∫(x³ + 1)/(x + 1) dx?

  1. Use u-substitution
  2. Use polynomial long division first
  3. Use partial fractions
  4. Cannot be integrated
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The correct answer is B. When the numerator degree ≥ denominator degree, use long division first. x³ + 1 = (x + 1)(x² − x + 1), so the fraction simplifies to x² − x + 1, which integrates easily.

Concept Tested: Long Division Method

10. What is ∫₀^π sin(x) dx?

  1. 0
  2. 1
  3. 2
  4. π
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The correct answer is C. F(x) = −cos(x). ∫₀^π sin(x) dx = [−cos(x)]₀^π = −cos(π) − (−cos(0)) = −(−1) − (−1) = 1 + 1 = 2.

Concept Tested: Integral Properties