Try:

FullSimplify[
 Integrate[(1/(s*Sqrt[2*Pi]))*Exp[-1/2*(((x - m)/s)^2)], x], 
 s \[Element] Reals \[And] s > 0 \[And] x \[Element] Reals \[And] 
  m \[Element] Reals]

(*-(1/2) Erf[(m - x)/(Sqrt[2] s)]*)

Integrate[(1/(s*Sqrt[2*Pi]))*Exp[-1/2*(((x - m)/s)^2)], {x, -Infinity, Infinity}, Assumptions -> {s > 0, m > 0}]
(* 1 *)

Integrate[(1/(s*Sqrt[2*Pi]))*Exp[-1/2*(((x - m)/s)^2)], {x, 0, Infinity}, Assumptions -> {s > 0, m > 0}]
 (*1/2 (1 + Erf[m/(Sqrt[2] s)])*)

Looks different, but answer is the same.

Try:

Integrate[
 x*(1/(s*Sqrt[2*Pi]))*Exp[-0.5*(((x - m)/s)^2)], {x, -Infinity, 
  Infinity}, Assumptions -> {s > 0, m > 0}]
f[m_] := m; Table[
Plot[{Evaluate[INT] // Re, f[m]}, {m, 0.01, 10}, 
PlotStyle -> {Black, {Dashed, Red}}, 
PlotLegends -> {"integral with 0.5 not 1/2", "f[m]"}], {s, 1, 10, 1/2}]

(*s from: 1 to 10 and m from: 0.01 to 10*)