Math Indefinite integrals Constant of integration

# Constant of integration

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

$F(x)+C$

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### Remember

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

### 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 $F'(x)=x^2=f(x)$
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### 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.