🚀 go-pugleaf

I am trying to determine the coefficients of a nonlinear
function which fit a probability distribution (x,P(x)). The
distribution P(x) is a set of data points (usually several
thousand) generated by a computer model and thus contains
noise. 

I have discovered that it is not only possibly to fit the
function directly to the data, but also it is possible to
fit the integral of the function to the integral of the
data.  In fact, the integral of the data can be a smoother
function. 

I would like to know if examining the "fitted integral" in
this way can be a useful technique for determining the
"best-fit" coefficients.  Although one must be careful about
what one means by "best-fit" and "goodness of fit", it seems
to me that examining both the function and its integral
might give more information than by examining the function
alone.  For example, it is conceivable that the data set as
well as its integral may both have physical meaning.
However, it is also conceivable that the integral of the
data contains less information than the data itself.

I have no background in statistics and thus any
help/comments/suggestions/pointers-to-the-literature would
be _greatly_ appreciated!