Problems with falign* environment

If I've well understood what you want, it can be obtained with align* nested in a fleqn environment from nccmath:

documentclass{article}
usepackage{lipsum}
usepackage{amsmath, amssymb}
usepackage{nccmath}

begin{document}

lipsum[11]
begin{fleqn}
begin{align*}
int_{S_{min}}^{infty} S  S^{-frac{5}{2}}  dS
  &= int_{S_{min}}^{infty} S^{-frac{3}{2}}  dS 
  &propto bigg[S^{-frac{1}{2}}bigg]^{S_{min}}_{infty} 
  &propto dfrac{1}{sqrt{S_{min}}}
end{align*}
end{fleqn}

end{document} 

I'm not sure where you got falign* from, but it's not a standard environment. You possibly mean flalign*.

There must be no blank line inside math displays. And you don't want split inside, but at least a line with three alignment points, in order to achieve left alignment.

documentclass{article}
usepackage{amsmath}

begin{document}

Some text before the display
some text before the display
some text before the display
some text before the display
begin{flalign*}
int_{S_{min}}^{infty} S , S^{-frac{5}{2}} , dS
  &= int_{S_{min}}^{infty} S^{-frac{3}{2}} , dS 
  &propto biggl[S^{-frac{1}{2}}biggr]^{S_{min}}_{infty} 
  &propto dfrac{1}{sqrt{S_{min}}}
&&
end{flalign*}
and some text after it.

end{document}

On the other hand, I see no reason for left alignment.

Note min rather than min. It “should” be mathrm{min}, but the final result is the same. Using , is preferable to “backslash-space”; biggl (left) and biggr (right) should be used rather than the unadorned bigg.