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Introduction to Integrals · Ages 17–18

Antiderivatives and Indefinite Integrals Practice

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Intended for Grade 12 students. Find each indefinite integral term by term, remember the arbitrary constant, and check your answers by differentiating.

  1. Find the antiderivative of f(x) = 7.

  2. Check by differentiating whether F(x) = x^3 - 5x + 4 is an antiderivative of f(x) = 3x^2 - 5.

  3. Find the indefinite integral of f(x) = 6x^2.

  4. Find the antiderivative of f(x) = 4x^3 + 6x^2 - 2x + 9.

  5. Check whether F(x) = 2x^4 + 3x^2 is an antiderivative of f(x) = 8x^3 + 6x - 1.

  6. Find the indefinite integral of f(x) = 15x^4 - 8x^3 + 3.

  7. Explain why the antiderivative of f(x) = 2x cannot simply be written as x^2.

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Antiderivatives and Indefinite Integrals Practice

Answer key

  1. Find the antiderivative of f(x) = 7.

    7x + C

  2. Check by differentiating whether F(x) = x^3 - 5x + 4 is an antiderivative of f(x) = 3x^2 - 5.

    Yes, because F'(x) = 3x^2 - 5, which matches f(x).

  3. Find the indefinite integral of f(x) = 6x^2.

    2x^3 + C

  4. Find the antiderivative of f(x) = 4x^3 + 6x^2 - 2x + 9.

    x^4 + 2x^3 - x^2 + 9x + C

  5. Check whether F(x) = 2x^4 + 3x^2 is an antiderivative of f(x) = 8x^3 + 6x - 1.

    No, because F'(x) = 8x^3 + 6x, which is missing the -1 term.

  6. Find the indefinite integral of f(x) = 15x^4 - 8x^3 + 3.

    3x^5 - 2x^4 + 3x + C

  7. Explain why the antiderivative of f(x) = 2x cannot simply be written as x^2.

    Writing only x^2 omits all other functions whose derivative is 2x, such as x^2 + 3 or x^2 - 9. The term + C is needed to include every valid antiderivative.