反常积分∫0到无穷e^(-x^2)dx=

二重积分的极坐标变换 ∫e^(-x²)dx=∫e^(-y²)dy 故(∫e^(-x²)dx)² =∫e^(-x²)dx∫e^(-y²)dy =∫∫e^[-(x²+y²)]dxdy =∫dθ∫e^(-r²)rdr =2π∫e^(-r²)rdr =-π∫e^(-r²)d(-r²) =-πe^(-r²)| =π 即∫e^(-x²)dx=√π