反常积分∫(0,正无穷)dx∫(x,根号3 x)e^-(x^2+y^2)dy

题目

反常积分∫(0,正无穷)dx∫(x,根号3 x)e^-(x^2+y^2)dy

答案

∫ dx ∫ (x,√3 x)e^[-(x^2+y^2)]dy
=∫ dt ∫ (0,+∞)e^(-r^2)rdr
=(π/12)∫ (0,+∞)(-1/2)e^(-r^2)rd(-r^2)
=(π/24)[-e^(-r^2)] (0,+∞) =π/24.