I don't know the syntax for Wolfram Alpha but in Mathematica:
In[1]:= NIntegrate[Sqrt[x] Exp[-(x - 1)^2], {x, 0, Infinity}]
NIntegrate[x^(-1/2)*Exp[-(x - 1)^2], {x, 0, Infinity}]
Out[1]= 1.64362
Out[2]= 1.97373
Added comment: I just now entered your equation into Wolfram Alpha and it does recognize your formula as you intended. Sorry, don't know what to say.