As a concept, the integration to infinity is a limit of the value of the integral with finite boundaries (I say boundaries, not limits, to avoid confusion).
However, the infinite integral can often be evaluated by a technique named 'contour integration' (about which the great physicist Richard Feynman wrote: 'One thing I never did learn was contour integration') for functions for which the finite integral would be terribe to do.
See 'Methods of contour integration' in Wikipedia. For this method you need some familiarity with the behavior of functions on the complex plane, residue theorem and the like.
If our aim is to evaluate a function f(x) in the extended real line , We use the concept of (minus infinity to plus infinity). This integral will give the measure [Sum of f(x) ] over the limit (minus infinity to plus infinity).
I suppose that both f_1 and f_2 polynomials are real functions (all coefficients of f_1(x) and f_2(x) are real). In such a case the requested integral may be calculated as a sum of integrals taken on sections of a real line ended on zeroes of the denominator f_2(x) and (-infinity, smallest zero of f_2] and [largest zero of_2, +infinity). All those partial results of integration have to be finite to get a meaningful result. If they are infinite, but of the same sign, then your result is infinite with the same sign. Otherwise the result is non-existent.
More generally, and maybe more in spirit of your original question: the integral from minus infinity to plus infinity is a limit of an integral from 'a' to 'b' taken when 'a' tends to minus infinity and 'b' tends to plus infinity independently. That is to say you shouldn't set 'a'=a0 - ct and 'b'=b0 + ct, where a0, b0 and c>0 are constants, and compute the above limit when t goes to infinity. Such a limit, if exists, is called "principal value" but is *not* the true integral you are looking for.
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