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Quiz: Basic Antiderivatives

Test your understanding of antiderivatives with these review questions.

1. An antiderivative of f(x) is a function F(x) such that:

  1. F(x) = f(x)
  2. F'(x) = f(x)
  3. F(x) = f'(x)
  4. F'(x) = f'(x)
Show Answer

The correct answer is B. An antiderivative F(x) of f(x) satisfies F'(x) = f(x). Finding antiderivatives is the reverse process of differentiation.

Concept Tested: Antiderivative

2. The notation ∫f(x) dx represents:

  1. The derivative of f
  2. The definite integral
  3. The indefinite integral (family of all antiderivatives)
  4. The limit of f
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The correct answer is C. The indefinite integral ∫f(x) dx represents the family of all antiderivatives of f(x), differing by an arbitrary constant C.

Concept Tested: Indefinite Integral

3. Why do we add "+ C" to indefinite integrals?

  1. C stands for "calculus"
  2. Functions differing by a constant have the same derivative
  3. To make the answer positive
  4. It's optional
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The correct answer is B. Since the derivative of any constant is zero, there are infinitely many antiderivatives differing by a constant. The "+C" represents all of them.

Concept Tested: Constant of Integration

4. What is ∫x⁴ dx?

  1. 4x³ + C
  2. x⁵/5 + C
  3. x⁵ + C
  4. 5x⁵ + C
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The correct answer is B. Using the Power Rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. So ∫x⁴ dx = x⁵/5 + C.

Concept Tested: Power Rule Integration

5. What is ∫cos(x) dx?

  1. −sin(x) + C
  2. sin(x) + C
  3. cos(x) + C
  4. −cos(x) + C
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The correct answer is B. Since d/dx[sin(x)] = cos(x), the antiderivative of cos(x) is sin(x) + C.

Concept Tested: Integral of Cos x

6. What is ∫sin(x) dx?

  1. cos(x) + C
  2. −cos(x) + C
  3. sin(x) + C
  4. −sin(x) + C
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The correct answer is B. Since d/dx[−cos(x)] = sin(x), the antiderivative of sin(x) is −cos(x) + C.

Concept Tested: Integral of Sin x

7. What is ∫(3x² + 2x − 1) dx?

  1. 6x + 2 + C
  2. x³ + x² − x + C
  3. 3x³ + 2x² − x + C
  4. x³ + x² + C
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The correct answer is B. Integrate term by term: ∫3x² dx = x³, ∫2x dx = x², ∫(−1) dx = −x. Combined: x³ + x² − x + C.

Concept Tested: Sum Rule Integration

8. What is ∫5 dx?

  1. 5
  2. 5x + C
  3. 0
  4. C
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The correct answer is B. The integral of a constant k is kx + C. Here, ∫5 dx = 5x + C.

Concept Tested: Constant Rule Integration

9. What is ∫sec²(x) dx?

  1. sec(x)tan(x) + C
  2. tan(x) + C
  3. −cot(x) + C
  4. 2sec(x) + C
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The correct answer is B. Since d/dx[tan(x)] = sec²(x), the antiderivative of sec²(x) is tan(x) + C.

Concept Tested: Integral of Sec Squared

10. If F'(x) = f(x) and F(0) = 5, this type of problem is called:

  1. A definite integral
  2. An initial value problem
  3. An optimization problem
  4. A limit problem
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The correct answer is B. An initial value problem (IVP) gives a differential equation and a condition to determine the particular antiderivative. Here, find F(x), then use F(0) = 5 to find C.

Concept Tested: Particular Antiderivative