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Evaluating integral with varying upper limit?


Given a function  f[x]  which happens to be rather messy and not 
analytically integrable, I want to evaluate the function

   g[y_] := NIntegrate[f[x], {x, ymin, y} ]

with ymin fixed and  ymin < y < Infinity.  

I suppose that FunctionInterpolate is the way to go here (???).

But, are there tricks to tell FunctionInterpolate what I know in 
advance, namely that f[x] is everywhere positive, and decreases toward 
zero rapidly enough at large x that g[y] will approach a finite limiting 
value as y -> Infinity? (which value I'd like to have FI obtain with 
moderate accuracy -- meaning 3 or 4 significant digits, not 10 or 20)

Thanks . . .


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