How to notate higher anti-derivatives? We can represent the $nth$ derivative of $y$ with the following notation:
$$\frac{d^ny}{dx^n}$$
How can we represent the $nth$ anti-derivative of $y$?
 A: To put a bit more context around this question, let's consider second derivatives.
The notation $\frac{d^2y}{dx^2}$ is only one of at least four widely-used
notations for representing such a derivative:
$$\frac{d^2 y}{dx^2},\quad  y'',\quad  \ddot y,\quad  \mbox{or}\quad  D^2 y.$$
The notations $y''$ and $\ddot y$ depend on an implicit understanding of what variable
we are to differentiate over;
for example, in many contexts $\ddot y$ is defined as a second derivative
with respect to time.
A variation on $D^2 y$ is to write $D_x^2 y$ so as to make that variable explicit.
Perhaps you might consider $D_x^2 y$ a fifth notation.
For higher derivatives there is another variation,
for example, one might write successive derivatives of $y$ as
$y'$, $y''$, $y'''$, $y^{(4)}$, $y^{(5)}$, and so forth.
You could say there is yet another type of notation (exemplified by $y^{(5)}$)
in this list, since you might sometimes see
$y^{(3)}$ instead of $y'''$ or $y''''$ instead of $y^{(4)}$,
although one rarely sees a second derivative written $y^{(2)}$.
Only some of these notations generalize well to arbitrary $n$th derivatives:
$$\frac{d^n y}{dx^n},\quad y^{(n)},\quad D^n y,\quad \mbox{or}\quad D_x^n y.$$
As suggested in one comment, for antiderivatives you could write
$$\frac{d^{-n}y}{dx^{-n}},\quad y^{(-n)},\quad D^{-n}y,\quad \mbox{or}\quad D_x^{-n}y.$$
Alternatively, as suggested in another comment, make an operator $I$ corresponding
to a kind of inverse of the operator $D$, and write
$$I^n y\quad \mbox{or}\quad I_x^n y.$$
A wrinkle is it is not strictly correct to write $I_x D_x y = y.$
Instead, $I_x D_x y$ is a parameterized family of functions $y + a_0$,
where $a_0$ is the parameter.
It follows that $I_x^n D_x^n y = y + P(x)$ where $P$ is a polynomial of
degree $n - 1$, that is, the $n$th antiderivative is a family of functions
parameterized by the $n$ coefficients of $P$.
On the other hand, $D_x^n I_x^n y = y$ without the need to introduce parameters.
