Symbolic Methodology

[Orion1]

Equation1:
$$\frac{d^2}{dx^2} (x^n) = \frac{d}{dx} \left[ \frac{d}{dx} (x^n) \right]$$

The LHS for Equation1 is the symbolic condensed version for the RHS, however, what is the LHS symbolic condensed version for Equation2 RHS?

Equation2:
$$\text{?} = \int \left[ \int \left( x^n dx \right) \right] \; dx$$

 

[Maxos]

$$\int dx \int \left( x^n dx \right)$$

 

[Orion1]

$$\int dx \int \left( x^n dx \right)$$

Interesting, I have never seen that version before. I was expecting something as:
$$\int \int x^n dx dx = \int \left[ \int \left( x^n dx \right) \right] \; dx$$

However, what if I wanted to demonstrate an equation that must be integrated 10 times or even 100 times? Surely there must be a shorthand version?

 

[Hurkyl]

It's somewhat rare to see iterated indefinite integrals: generally you would specify bounds, even if it's something like:

$$\int_0^x \int_0^t f(s) \, ds \, dt$$

I've often seen high dimensional integrals written something like:

$$\iint \cdots \int f(x_1, \ldots \, x_n) \, dx_1 \, dx_2 \, \cdots \, dx_n$$

with some additional text indicating the region of integration... or instead written as a single integral whose dummy variable ranges over a high-dimensional space.


Another option, which I suspect is the best one for you, is to define an integral operator. For example, you could define the operator I via:

$$(If)(x) := \int_0^x f(t) \, dt$$

and then you could indicate an iterated integral by $I^nf$.

 

[iNCREDiBLE]

You can write $$D^{-2}f(x)$$ and/or $$D^{-2}(x^n)$$.