Integral in $[\epsilon,\infty]$, no anal. expr. but can approx. in $[\epsilon,x\rightarrow\epsilon]$ and $[y\rightarrow \infty,\infty]$. Best approx?

Question

Say I have an integral in $[\epsilon,+\infty]$, with $\epsilon>0$, that does not allow for analytical expressions, but I can approximate its value in $[\epsilon,x]$ and in $[y,+\infty]$ for $x\rightarrow \epsilon$ (this one is $k*ln(x/\epsilon)$ with $k$ a known constant) and $y\rightarrow +\infty$ (this one is $k'e^{-y^{2}}/y^{5}$ with $k'$ a known constant). Which would be the best way to approximate it?

Answer 1

Best where? One approximation is good near $\epsilon$, another is good for very large $x$; between those extremes, one or both or neither might be good. It's impossible to say from this distance, but I suspect you might be better off with an approximation method that isn't based on either of those limits: perhaps one of the standard numerical integration methods.