# Constant of integration

A function with an antiderivative $F(x)$ has infinitely more antiderivatives. All of the form:

$F(x)+C$

!

### Remember

The

**constant of integration**$C$ is an arbitrary real number ($C\in\mathbb{R}$) and is important for indefinite integrals.i

### Explanation

When taking the derivative, the constant $C$ would disappear (constant rule).

Regardless of $C$ all functions therefore convert to the initial function when taking the derivative, which corresponds to the definition of antiderivatives.

Regardless of $C$ all functions therefore convert to the initial function when taking the derivative, which corresponds to the definition of antiderivatives.

### Example

Set up three antiderivatives of $f(x)=x^2$.

- $F(x)=\frac13x^3$, since $F'(x)=x^2=f(x)$
- $F(x)=\frac13x^3+2$, since $F'(x)=x^2=f(x)$
- $F(x)=\frac13x^3-4$, since $F'(x)=x^2=f(x)$

i

### Hint

Since the constant of integration is any number, adding multiple constants of integration ($C_1+C_2+C_3=C$) results in another arbitrary number, which can also be seen again as a constant of integration.

In order to save paperwork, one usually adds $C$ only at the end when calculating with indefinite integrals.

In order to save paperwork, one usually adds $C$ only at the end when calculating with indefinite integrals.