Question

求教一个积分问题,下面的积分是怎么计算出来的?

Answer 1

需要利用二重积分来证明.

[∫e^(-t^2)dt]^2 = ∫e^(-x^2)dx∫e^(-y^2)dy

=∫∫e^-(x^2+y^2)dxdy

=∫∫e^-r^2*rdrdθ

=∫dθ∫e^(-r^2)rdr

=2π * (-1/2)∫e^(-r^2)d(-r^2)

=-π * [e^(-∞) - e^0]

=π

所以,∫e^(-t^2)dt = √π