Originally Posted by
Drexel28 That's because you've never heard the real definition. I don't mean this in a mocking way, it's just the truth. The Fundamental Theorem of Calculus, which is surely what you refer to, states (roughly) that if $\displaystyle f=F'$ then $\displaystyle \displaystyle \int_a^b f(x)\text{ }dx=F(b)-F(a)$. But, what if $\displaystyle f$ doesn't satisfy the hypotheses? The definition of integrability etc. is kind of 'high-tech' and 'notation laden'. See Apostol or Rudin for a full explanation.