Suppose \(f\) is continous on an interval \(I\) and \(a \in I\). For all \(x \in I\) let

\(F(x) = \int_a^x f(t) \mathrm{d}t\)

Then \(F\) is an antiderivative of \(f\) on \(I\).

Suppose \(f\) is continous on an interval \(I\). Then \(f\) has an antiderivative on \(I\).

Suppose \(f\) is a function that is continous on an interval \(I\) and \(F\) is an antiderivative of \(f\) on \(I\). Then for any numbers \(a\) and \(b\) in \(I\),

\(\int_a^b f(x) \mathrm{d}x = F(b) - F(a)\)