Evaluate the integral from 0 to 1 (x * e^ (-x^2) dx)

I know the answer is .31606 , I just don't know how to get it

... ∫ [ x. (e^(-x²)) dx ........................ on [0,1]

= (-1/2) ∫ (e^(-x²))· (-2x) dx

= (-1/2) ∫ (e^u) du, ............... u = -x², du = (-2x) dx

= (-1/2) [ (e^u) ]

= (-1/2) [ e^(-x²) ] .................. on [0,1]

= (-1/2) [ (e^(-1)) - (e^0) ]

= (-1/2) [ (1/e) - 1 ]

= (1/2)[ 1 - (1/e) ]

= 0.3160602794 ........................................… Ans.
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