广义积分∫(0,+∞)e^x^2dx=√π/2怎么求的?

Question

广义积分∫(0,+∞)e^x^2dx=√π/2怎么求的?

Answer 1

说明：此题少打了一个负号!应该是“广义积分∫(0,+∞)e^(-x^2)dx=√π/2怎么求的?”才对.
设∫(0,+∞)e^(-x²)dx=T
∵T²=[∫(0,+∞)e^(-x²)dx]*[∫(0,+∞)e^(-x²)dx]
=[∫(0,+∞)e^(-x²)dx]*[∫(0,+∞)e^(-y²)dy]
=∫∫(D)e^(-x²-y²)dxdy (积分区域D：0≤x≤+∞,0≤y≤+∞)
=∫(0,π/2)dθ∫(0,+∞)e^(-r²)rdr (极坐标变换)
=(π/2)*[-e^(-r²)/2]│(0,+∞)
=(π/2)*(-0+1)
=π/2
∴T=√(π/2)
故∫(0,+∞)e^(-x²)dx=√(π/2).