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
The indefinite integral as the function depending on any \(x\in I\):
\[F(x):=\int_a^x f(t)dt.\]
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