Integrate[..., {x, -infinity, infinity}] gives a result while integral does not converge

Questions
1

Integrate[1/x^2 Exp[- x^2/(2 q)]/Sqrt[2 [Pi] q], {x, -infinity, infinity}]

gives

-(1/q)

While, in fact, the integral does not converge…

Version 12.3.1.0

2

If we treat function as continuous one at {-infinity, infinty} and with Regularization then:

Integrate[1/x^2 Exp[-x^2/(2 q)]/Sqrt[2 \[Pi] q] x^s, {x, -Infinity, Infinity},
        Assumptions -> {q > 0, s >= 0}, PrincipalValue -> True]

(*(2^(-2 + s/2) (1 + (-1)^s) q^(-1 + s/2)*Gamma[1/2 (-1 + s)])/Sqrt[\[Pi]]*)

Limit[%, s -> 0]

(*-(1/q)*)

Maybe is a bug ,maybe not. Send this example to WRI you will find out what they have to say about it.

3

The function is positive when q > 0, and the output is negative: the result is clearly wrong. It seems to be ignoring the singularity at x == 0 and doing something like this:

f[x_, q_] = Integrate[1/x^2 Exp[-x^2/(2 q)]/Sqrt[2 \[Pi] q], x]
Limit[f[x, q], x -> Infinity] - Limit[f[x, q], x -> -Infinity]

When the parameter is numerical, Mathematica detects the singularity:

With[{q = 1},
 Integrate[1/x^2 Exp[-x^2/(2 q)]/Sqrt[2 \[Pi] q],
  {x, -Infinity, Infinity}]]
4

This is also interesting:

Table[{n, Integrate[1/x^n Exp[-q x^2],
   {x, -Infinity, Infinity}]},
 {n, 6}]