Integrating a term again and again

Question

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to $f(x) = 4x^3$, it is now still really wanting that last integral.

What is the notation for an integral above the function:

$\int 4x^3 dx=x^4$, but is there notation relevant to this, and what is the meaning/purpose of such a construct

After much time, it seems no such notation exists, is this correct? As suggested by the answer below, it may be related to the lack of need for such thing.

Answer 1

$$f''(x)=g(x)$$

is simply a differantial equation, most of them arise naturally from physics. As an example

if $X(t)$ the position of a particule then acceleration $a(t)=X''(t)$.

If we just know the acceleration, we can find $X(t)$ (almost) by integrating twice.

There is no standart "notation" for integrating "$n$ times".

Answer 2

Hmm, all I've seen is this sort of thing $\underbrace{\int\int\cdots\int}_{n} f(x) ~ dx\cdots dx$.

Answer 3

From what (little) I've seen of fractional calculus, there seems to be a convention to take $$J^nf(t) = \underbrace{\int\cdots\int_0^t}_n f(x) ~dx \cdots dx.$$

I'm not sure if I'd call it standard (I haven't seen many applications where 'higher order' integrals are even used). It does have the property that it is uniquely defined (because of the definite integration).

Answer 4

To add to the present list of answers, I have seen the notation

$$\frac{d^{-n} f(x)}{dx^{-n}},$$

used once on a very exceptional occasion, though there exists no standard notation to describe the $n$th anti-derivative (as already mentioned).