Can an indefinite double integral of a single variable be defined? The idea is basically integrating the result of the indefinite integral to the same variable.
$$ \int \left(\int f(x)dx \right)dx $$ Edit: Forgot to ask it at first, would the following be a correct for it as well? $$\iint f(x) dx^2 $$
The expression $\displaystyle \int f(x) \, dx$ is typically understood to mean "a general antiderivative of $f$". Any two such antiderivatives (on an interval) only differ by a constant.
It would be entirely consistent for your notation to be read as "a general antiderivative of a general antiderivative of $f$". As such, you could write
$$\int \left( \int x^2 \, dx \right) \, dx = \int \frac {x^3}{3} + C \, dx = \frac{x^4}{12} + Cx + D.$$
This (in my experience) is highly nonstandard, however.
Sure it can. For an easy example, take any function $g(x)$ whose second derivative $g''(x)$ exists.
Then we have $$\int \left(\int g''(x) dx\right)dx $$ $$ = \int (g'(x)+c_1) dx$$ $$ = g(x) +c_1x+c_2$$
For constants $c_1$ and $c_2$
For your second question, I am not personally familiar with the notation of $dx^2$ but it appears to be used for solving problems such as How to integrate with respect to $x^2$?
