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In[]:=
Integrate[(x^2*(Erf[(T + x)/Sqrt[2]] + Erf[(T - x)/Sqrt[2]]))/E^(x^2/(2*y^2)), {x, 0, Infinity}]
Integrate[(x^2*(Erf[(T+x)/Sqrt[2]]+Erf[(T-x)/Sqrt[2]]))/E^(x^2/(2*y^2)),{x,0,Infinity}]
Out[]=
∞
∫
0
-
2
x
2
2
y

2
x
Erf
T-x
2
+Erf
T+x
2
x