Question

怎么求E的负X平方次方在负无穷到正无穷间的广义积分

Answer 1

从0到正笑山桥无穷大x*x*（e的负（x的平方））唯举
=∫(x^2)*e^(-x^2)dx
=(∫x*e(-x^2)dx^2)/2
=-(∫x
d(e^(-x^2)))/2
=-x*e^(-x^2)/2+(∫e^(-x^2)
dx)/2
=0+(∫e^(-x^2)
dx)/2
令t=(∫e^(-x^2)
dx)/2=(∫e^(-y^2)
dy)/2
t*t=((∫e^(-x^2)
dx)/2)*((∫e^(-y^2)
dy)/2)
=∫∫e^(-x^2-y^2)dxdy/4
接下来换元令x=rcosθ，y=rsinθ
可得到积分碰猛的结果

Answer 2

I=[∫e^(-x^2)dx]*[∫e^(-y^2)dy]
=∫∫e^(-x^2-y^2)dxdy
转化成极坐标
=[∫蔽乱(0-2π)da][∫(0-+无穷）派并枯e^(-p^2)pdp]
=2π*[(-1/2)e^(-p^2)|(0-+无穷尘洞)]
=2π*1/2
=π
∫e^(-x^2)dx=I^(1/2)=√π