Question

高数 这一步怎么算的？

Answer 1

u/√2=x/β
du = (√2/β) dx
=[-βxe^(-x^2/β^2)]|(0->+∞) +β∫(0->+∞) e^(-x^2/β^2) dx

=0 +β∫(0->+∞) e^(-x^2/β^2) dx
=0 +(β^2/√2 )∫(0->+∞) e^(-u^2/2) du
=[ β^2.√(2π)/√2 ]∫(0->+∞) [1/√(2π)] e^(-u^2/2) du
=β^2.√π .∫(0->+∞) [1/√(2π)] e^(-u^2/2) du
=β^2.√π . (1/2)
=β^2.√π /2
//
X~N(0,1)
f(x) = [1/√(2π)] e^(-x^2/2) ; -∞<x< +∞
∫(-∞->+∞) [1/√(2π)] e^(-x^2/2) dx =1
=>
∫(0->+∞) [1/√(2π)] e^(-x^2/2) dx =1/2