∫e^(-x^2)dx=∫e^(-y^2)dy
而∫e^(-x^2)dx*∫e^(-y^2)dy
=∫∫e^(-y^2)*e^(-x^2)dxdy
=∫∫e^(-x^2-y^2)dxdy
然后是用
极坐标换元,x=rcosa,y=rsina
r属于[0,
无穷大),a属于[0,2π]
=∫∫re^(-r^2)drda
(r属于[0,无穷大),a属于[0,2π])
=∫(0,2π)da*∫re^(-r^2)dr
r属于[0,无穷大),
=2π*
1/2*∫e^(-r^2)dr^2
r属于[0,无穷大),
=π*
∫-de^(-r^2)
r属于[0,无穷大),
=π*[e^(-0^2)-lime^(-r^2)]
r→无穷大
=π*(1-0)
=π
∫e^(-x^2)dx*∫e^(-y^2)dy=π=[∫e^(-x^2)dx]^2
易知∫e^(-x^2)dx>0
所以∫e^(-x^2)dx=√π