Symbolic Methodology...

Orion1

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

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

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

Maxos

[tex]\int dx \int \left( x^n dx \right)[/tex]

Orion1

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

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:

[tex]
\int_0^x \int_0^t f(s) \, ds \, dt
[/tex]

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

[tex]
\iint \cdots \int f(x_1, \ldots \, x_n) \, dx_1 \, dx_2 \, \cdots \, dx_n
[/tex]

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:

[tex](If)(x) := \int_0^x f(t) \, dt[/tex]

and then you could indicate an iterated integral by [itex]I^nf[/itex].

iNCREDiBLE

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