Question

How do we denote the nth antiderivative of a function?

Given that we have some function, say `f(x)=5x^3+3x^2-3`, the first derivative would be denoted as `f'(x)=15x^2+6x` or `f^1(x)` ...
I know the first antiderivative of `f(x)` would be written as `F(x)`. But do we have anything to show the second and third antiderivatives?

Answer 1

Capitalization, going from function `f` to antiderivative `F`, is usually only used until the indefinite integral and its notation are introduced.

Explanation:

In some treatments (for example Calculus by James Stewart) , there is a difference between the "most general antiderivative" and "the indefinite integral".
I am not aware of a standard notation for the such 'higher' antiderivatives.

For integrals, we use `int f(x) dx = F(x) +C` where `F'(x) = f(x)`

and the we can (if we wish) repeat:

`int int f(x)dx dx = G(x) +C_1x+C_2` where `G''(x) = f(x)`

If you are interested in exploring 'higher order' antiderivatives, I suggest the notation `f^((-1))(x) +C` for the first antiderivative and `f^((-2))(x) +C_1x + C_2` for the second and so on.
With this notation you have `f^((0))(x) = f(x)` and `f^((1))(x) = f'(x)` and so on.