Date: Around 1993
Does the integral of x^2(dx)^.5 have any meaning? If it does,
how does one go about integrating the expression?
I am not familiar with any meaning for this, but if we change it
slightly to x^2*d(x^.5), then substitution methods do apply, etc. In the usual
sense of the definition of differentials, (dx)^.5 will never appear, so
something other than Riemannian Integration would have to be defined to give
this a foundation.
There is something called "fractional integration," which I
remember hearing about many years ago. You could try and find a book on this.
It is based on a relation between multiple integration of one-dimensional
functions and the B-function (sorry, I forgot the proper name). Anyway, I do
not recall that fractional integrals were actually very useful.
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Update: June 2012