Question

What is the difference between the corollary and the second part of the Fundamental Theorem of Calculus?

Answer 1

In some treatments it is simply a matter of naming.

Explanation:

Some presentations use different names for exactly the same two statements.

In other presentations: the corollary assumes that the integrand is continuous while Part 2 assumes that the integrand is integrable (and has an antiderivative).

Since every continuous function is integrable, but NOT conversely, the second alternative makes part 2 a stronger statement than the corollary.