What is the difference between Definite Integral & Indefinite Integral on the basis of their connection with derivatives?

Question

I'm reading Denlinger's text on Real Analysis. There is a difference mentioned between definite integrals and indefinite integrals:

The definite integral is a fundamentally significant concept, existing independently of any connection with derivatives. The 'indefinite' integral $ \int f(t) dt $, on the other hand, depends upon the notion of derivatives for its definition

But, I think both 'definite' integral and 'indefinite' integral have their connection with derivatives as is clear from Fundamental theorem of calculus. May be i am not fully getting what the above text wants to convey.

Please, explain this text in detail & help me in sorting out my confusion.

Another question: Is definite integral is also called Riemann Integral?

Answer 1

Definite integrals are defined only using the concept of Riemann sum, therefore you don't need derivatives in order to talk about them. Ah, if you want to compute them - this is an entirely different business! Nobody computes definite integrals using their definition, it would be too cumbersome, but using various other techniques, in particular the Fundamental Theorem of Calculus (which, indeed, needs derivatives). But don't mistake "definition" for "computation"!

Indefinite integrals, on the other hand, depend essentially on the concept of "derivative": $F$ is an indefinite integral of $f$ (denoted $F (x) = \int f(x) \ \Bbb d x$) if and only if $F' = f$.

Yes, if the interval on which you integrate is bounded and the function is bounded too, then "definite integral" is the same thing as "Riemann integral". If either the interval or the function is unbounded, you deal with an "improper integral" (which is not a Riemann integral, but is defined using a limit of Riemann integrals - but this is a different story).

Answer 2

The definite integral is by definition a limit of a certain class of sums (the class of the Riemann sums, by the way): no relation a priori with the derivatives. The indefinite integral is by definition the primitive of a function, or its 'antiderivative' (hence it is related, by definition, to the concept of derivative). So the two integrals are independently defined concepts. The Foundamental Theorem of Calculus establishes an amazing relation between these two independent concepts (because of that it is 'foundamental'). But this is a theorem, not a definition of integral: it establishes a connection which goes beyond to what the definitions of definite and indefinite integrals explicitely state.