information:filtering, linguistic:voice_change, structural:expansion, structural:scholar_edition
First part

This part is sometimes referred to as the first fundamental theorem of calculus.

Let f be a continuous real-valued function defined on a closed interval. Let F be the function defined, for all x in, by F (x) = ∫ a x f (t) d t. {\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.}

Then F is uniformly continuous on and differentiable on the open interval (a, b), and F ′ (x) = f (x) {\displaystyle F'(x)=f(x)} for all x in (a, b) so F is an antiderivative of f.