# Antiderivative

A function F whose derivative is f. Because constants vanish when you differentiate, every antiderivative comes with + C.

Maths (university) · Part 05: Integration · Card 1 of 10 · Last updated 2026-10-19

## Analogy

Rewinding a film: the derivative tells you the speed, the antiderivative rebuilds the journey.

## Example

**Antiderivative of x^2**

Try F(x) = x^3/3.
Check: F'(x) = 3x^2/3 = x^2.
So the integral of x^2 dx = x^3/3 + C.
C is any constant.

## In practice

Used to recover position from velocity or total cost from a rate.

## Practice

### Flashcards

- **What is the integral of x^2 dx?** x^3/3 + C.

## Questions people ask

### Why do we add + C to an indefinite integral?

Many functions share the same derivative, because constants disappear when you differentiate. x squared, x squared plus 5 and x squared minus 3 all give 2x. The C stands for every possible constant, so the answer covers the whole family of antiderivatives.

### How can I check an antiderivative is right?

Differentiate your answer. If you get the original function back, the antiderivative is correct. For example, if you claim the integral of x squared is x cubed over 3, differentiating gives x squared. This quick check catches most mistakes, including a forgotten power or a wrong denominator.

## Related concepts

- [Power rule for integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-2)
- [Riemann sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-3)
- [Definite integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-4)
- [Fundamental theorem of calculus](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-5)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-5-concept-1