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## Theorem: Indefinite Integral, Antiderivative

Let $$I$$ be a real interval (it could be closed, open, semi-open, or even unbounded). Let $$f:I\mapsto\mathbb R$$ be a continuous real function. Further, let $$[a,x]\subseteq I$$ be a closed interval contained in $$I$$. Because the continuity of $$f$$ on the closed interval $$[a,x]$$ is a sufficient condition for $$f$$ to be Riemann integrable on that interval, this motivates the following

### Definition

The indefinite integral as the function depending on any $$x\in I$$:

$F(x):=\int_a^x f(t)dt.$

### Theorem

The function $$F:I\mapsto\mathbb R$$ is differentiable and its derivative is
$F’=f.$
Because of this property, the indefinite integral $$F$$ is sometimes also called the antiderivative of $$f$$

| | | | | created: 2016-03-06 09:18:51 | modified: 2016-03-08 23:19:18 | by: bookofproofs | references: [581]

## 2.Proposition: Antiderivatives are Uniquely Defined Up to a Constant

(none)

### Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983