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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 \(f\)