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?

1 Answer
Feb 25, 2018

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.