I have often read that the Fundamental Theorem of Calculus (FTC) tells us that integration is the opposite of differentiation. I have always found this summary confusing, so I will lay out what I think people mean when they make such a statement.
The First FTC implies the existence of antiderivatives for every function, $f$, that is continuous on a particular interval, say $[a,b]$. Generally, we denote this antiderivative as $F$. Differentiating $F$ gets back to our original function, $f$. So when people say that 'integration is the opposite of differentiation', what they mean is that an antiderivative of a function can be computed using a definite integral.
The Second FTC is more powerful than the First FTC, as it tells us that definite integrals can be computed using the antiderivative of a function (which is generally more useful than knowing that one possible antiderivative of $f$ can be computed using a definite integral, $F$). For the Second FTC, I don't understand how this is related to 'integration being the opposite of differentiation' at all. The Second FTC shows us the link between antiderivatives (indefinite integrals) and definite integrals. It is extremely useful for trying to find the area under a curve, but I'm not sure how this relates to integration and differentiation being 'opposites'.
Is there something about the First FTC or the Second FTC that has a bigger implication about integration being the opposite of differentiation, or is my understanding correct?