9、计算∫（上下限0～+∞） xe^(-x^2) dx=
问题描述：
9、计算∫（上下限0～+∞） xe^(-x^2) dx=
答
[e^(-x^2)]' = e^(-x^2) * (-x^2)' = e^(-x^2) * (-2x) = -2 x * e^(-x^2)
∫x*e^(-x^2) dx = (-1/2) ∫ -2x * e^(-x^2) dx = (-1/2)∫d[e^(-x^2)]
= -(1/2)*e^(-x^2)
= -(1/2)*e^(-∞) - [(-1/2)*e^0]
= 0 + 1/2
= 1/2
答
∫xe^(-x^2)dx|(0,+∞)u=-x^2du=-2xdx∫xe^(-x^2)dx=-(1/2)∫e^udu=-(1/2)e^u+C=-(1/2)e^(-x^2)+C=-1/2e^(x^2)+C∫xe^(-x^2)dx|(0,+∞)=[-1/2e^(+∞^2)+C]-[-1/2e^(0^2)+C]=1/2