Skip to main content
Mathovia

Math Practice

Antiderivatives Practice Problems

Rewrite each integrand as a sum of basic terms, antidifferentiate term by term, and add + C. Type C for the constant, and use parentheses for functions, as in sin(x) or ln(x). Check each answer by differentiating it.

  1. 1.

    Find .

    Show full solution

    By the power rule, .

  2. 2.

    Find .

    Show full solution

    Term by term: .

  3. 3.

    Find .

    Show full solution

    The antiderivative of a constant is , so .

  4. 4.

    Find .

    Show full solution

    .

  5. 5.

    Find .

    Show full solution

    Write . Then .

  6. 6.

    Find .

    Show full solution

    Write . Then .

  7. 7.

    Find . Type ln(x) or ln|x| for the log.

    Show full solution

    (not the power rule, since ), and . So the answer is .

  8. 8.

    Find . Use sin(x) and cos(x).

    Show full solution

    and . So .

  9. 9.

    Find .

    Show full solution

    Split the fraction: . Then .

  10. 10.

    Find .

    Show full solution

    Multiply out first: . Then .

  11. 11.

    Find if and .

    Show full solution

    . Then , so and .

  12. 12.

    Find if , and .

    Show full solution

    Antidifferentiate twice. , and , so . Then , and , so .
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Antiderivatives Practice Problems

These problems follow the antiderivatives and indefinite integrals lesson. Each one asks for an antiderivative, or for the particular one that fits a given value.

The method

  1. Rewrite the integrand: roots as powers, fractions split, products multiplied out.
  2. Antidifferentiate term by term with the power rule, the 1/x rule and the trig and exponential rules.
  3. Add , or use the given value to find .
  4. Check by differentiating your answer.

Common mistakes to watch for

  • Forgetting on an indefinite integral.
  • Using the power rule on . Its antiderivative is .
  • Getting the sign of wrong. It’s .

Frequently Asked Questions

Do I have to type + C?+

Yes, for every indefinite integral. Use the letter C. The two initial-value problems at the end ask for one particular function, so they have no C.

How should I type my answers?+

Use ^ for powers and parentheses for function inputs: x^(3/2), sin(x), e^x, ln(x). Any equivalent form is accepted, so 0.5x^2 and x^2/2 both work.

More practice problems

Power Rule Practice Problems Derivatives of Trig Functions Practice Problems Derivatives of Exponential and Logarithmic Functions Practice Problems