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

Answer 1

∫ <0,+∞>dx ∫ (x,√3 x)e^[-(x^2+y^2)]dy
=∫ <π/4,π/3>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.