relationship between definite integral and derivative

[maxx]

I know that the indefinite integral and the derivatives are inverses of each other, but how about the definite integral?

 

[daon]

In Calculus, the definite integral is essentially a constant number (usually an extended real number). There's really no way of preserving the presence of a function after evaluating the integral. If you take the derivative of a definite integral, you get zero.

 

[Deleted member 4993]

In Calculus, the definite integral is essentially a constant number (usually an extended real number). There's really no way of preserving the presence of a function after evaluating the integral. If you take the derivative of a definite integral, you get zero.

Limits of definite integral can be a variable or function of variable. In that case, you can take derivative of that integral.

 

[pka]

This may well be a wrong reading of the basic question.
But I took the question to be asking about how the antiderivative can be use in evaluating the definite integral.
If that is the question, the ponder this.
There are functions, f, that are themselves everywhere derivatives of other functions, that is g’=f, but yet f has no ordinary definite integral on [0,1].