integration – is the definite integral always equal F(b)-F(a)?

Consider, you have F'(x) = f(x) for all x from (a,b), where F(x) is antiderivative. Does it necessarily exist the definite integral on an interval (a,b) and is equal to F(b)-F(a)?

From condition we could conclude that f(x) is integrable and continuous. It seems that definite integral exists. And I have no example, when definite integral isn’t equal to F(b) – F(a). Please, can you provide an example for such situation and list the properties of a function f(x). I know correct answer for question(my second sentence) is “no”, but I want to understand it