x^2e^(-x^2) 從負無限大積到無限大

sum x^2 e^(-x^2) dx

= (1/2) sum x e^(-x^2) dx^2　　(2xdx = dx^2)

= (-1/2) sum x de^(-x^2)　　(e^(-x^2)dx^2 = -de^(-x^2))

= (-1/2) [ xe^(-x^2)|(-inf~+inf) - sum e^(-x^2) dx ]　　(分部積分)

xe^(-x^2)|(-inf~+inf)
= x/e^(x^2)|(-inf~+inf)
= 1/2xe^(x^2)|(-inf~+inf)　　(L'Hospital's rule)
= 0 - 0 = 0

sum e^(-x^2) dx|(-inf~inf) = √pi


原式 = (-1/2) [ 0 - √pi ] = (1/2)√pi　　OA O

因為x^2 exp(-2x)是偶函數﹐所以考慮0至∞再將結果乘2即可

∫ x^2 exp(-x^2) dx (0 -> ∞)= (-1/2) ∫ x d(exp(-x^2)) (0 -> ∞)= (-1/2)x exp(-x^2)|(0,∞) + (1/2)∫ exp(-x^2) dx (0 -> ∞)

因為 lim (x->∞) x exp(-x^2) = 0 = (1/2)∫ exp(-x^2) dx (0 -> ∞)= √π/4因此∫ x^2 exp(-2x) = √π/2 (-∞ -> ∞)

w=∫x^2e^(-x^2) 從負無限大積到無限大該怎麼做呢?Let y=x^2 => dy=2x*dx => dx=dy/2x=dy/2y^0.5w=∫y*e(-y)*dy/2y^0.5=-∫0.5*y^0.5*d[e(-y)]=-0.5*y^0.5*e(-y)+∫0.5*e(-y)*0.5*y^(-0.5)*dy<部份積分>=-0.5*y^0.5*e(-y)+∫0.25*e(-y)*y^(-0.5)*dy=-0.5*y^0.5*e(-y)+w/2So w/2=-0.5*y^0.5*e(-y)=> w=-y^0.5*e(-y)=-(x^2)^0.5*e(-x^2)=-x/e(x^2)=-a+ba=Limit<x->∞>x/e(x^2)=1/[2x*e(x^2)]<l'Hopital定理>=0b=Limit<x->-∞>x/e(x^2)=1/[2x*e(x^2)]=0So w=0