Summation of Higher Order Integrals Probably a noob question, but How is it the notation of the summation of Higher order Integrals .. I mean, in the derivative case we have:
$$ \frac{d f(x)}{dx} + \frac{d^2 f(x)}{dx^2} + ... + \frac{d^n f(x)}{dx^n} =   \sum_{m=1}^n  \frac{d^m f(x)}{dx^m}$$ 
In the case of Integrals how do I represent the same concept?
$$ \int f(x)dx +  \int \int f(x)f(x) dx dx + ...   $$
Since Integrals are anti-derivatives, I don't know if it is possible to do this:
$$ \frac{d^{-1} f(x)}{dx^{-1}} + \frac{d^{-2} f(x)}{dx^{-2}} + ... + \frac{d^{-n} f(x)}{dx^{-n}} =   \sum_{m=1}^n  \frac{d^{-m} f(x)}{dx^{-m}}$$
Is there a way to do the Summation of Higher order integrals using the integral notation .. Is not possible, right?
 A: If you want to use the standard Leibniz integral notation, a possible notation that would be easily recognizable without additional explanation might be
$$
\int f(x)\,\mathrm dx + \iint f(x)\,\mathrm dx + \iiint f(x)\,\mathrm dx + \cdots
+ \int \!\!\!\!\underbrace{\cdots}_{n \text{ times}\ \ \, }\!\!\!\!\int f(x)\,\mathrm dx = \sum_{n=1}^N\int \!\!\!\!\underbrace{\cdots}_{n \text{ times}\ \ \, }\!\!\!\!\int f(x)\,\mathrm dx^n
$$
However, this is somewhat messy, and fills up a lot of the page which can be distracting. If you are willing to adopt an alternative notation for anti-differentiation, this can be made much more concise.
For example, taking some inspiration from history, we can use Lagrange's Notation to write
$$
f^{(-1)}(x) +f^{(-2)}(x) + \cdots + f^{(-n)}(x) = \sum_{n=1}^N f^{(-n)}(x)
$$
Or alternatively, in Euler's notation, we have
$$
D^{(-1)}f(x) +D^{(-2)}f(x) + \cdots + D^{(-n)}f(x) = \sum_{n=1}^N D^{(-n)}f(x)
$$
For absolute brevity, you could even define your own differential operator, $L_n$, by
$$
L_n(f) = \int f(x)\,\mathrm dx + \iint f(x)\,\mathrm dx + \iiint f(x)\,\mathrm dx + \cdots
+ \int \!\!\!\!\underbrace{\cdots}_{n \text{ times}\ \ \, }\!\!\!\!\int f(x)\,\mathrm dx
$$
and then denote the entire sum simply by $L_n f$ where ever it might appear.
As always in mathematics, you are free to adopt whatever notation you think will more clearly convey your ideas to your reader.
