# 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 $$f$$

Proofs: 1

Definitions: 1
Proofs: 2 3
Propositions: 4
Subsections: 5
Theorems: 6

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983