---
tag: 03VHANA6ZK
kind: definition
name: Indefinite integral
msc: [26A36]
defines: 0WN1QXR7R3
uses: [0BW9N5CDSB]
sources:
  - url: https://en.wikipedia.org/wiki/Antiderivative
    locator: Uses and properties
---

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ have an [[antiderivative|0BW9N5CDSB]] $F$ on $I$.
The indefinite integral of $f$ is the set of all antiderivatives of $f$ on $I$.
It is written
$$\int f(x) \, dx = F(x) + C,$$
where $C$ stands for an arbitrary real constant.

## Remarks

The constant $C$ is the constant of integration.
For example, $\int \cos x \, dx = \sin x + C$.
Some texts use the name for the function $x \mapsto \int_a^x f(t) \, dt$, for a fixed $a$.

## Explanations

### L1 human

The indefinite integral collects all the antiderivatives of a function.
They differ only by a constant, so one antiderivative plus $C$ describes them all.