Question

高数 积分一题。急求高手啊！！！！

反常积分。下限是0，上限是+无穷大。被积函数是 e^(-x^2) 对x做积分。求解啊。大神！！！！！

Answer 1

这题要用重积分来求
令L=∫(0,∞ )e^(-x^2)dx
那么
∫(0,∞ )e^(-x^2)dx*∫(0,∞ )e^(-y^2)dy
=∫(0,∞ )∫(0,∞ )e^(-x^2-y^2)dxdy 做变换x=rcost y=rsint
=∫(0,π/2 )dt ∫(0,∞ )e^(-r^2)rdr
=π/2*1/2*∫(0,∞ )e^(-r^2) d(r^2)
=π/4*e^(-r^2) (0,∞)
=π/4
而∫(0,∞ )e^(-x^2)dx=∫(0,∞ )e^(-y^2)dy
所以∫(0,∞ )e^(-x^2)dx=√(π/4)=√π/2

Answer 2

∫【0，＋∞】e^(-x²)dx=√【∫【0，＋∞】e^(-x²)dx * ∫【0，＋∞】e^(-x²)dx】
=√【∫∫[0,＋∞]×[0,＋∞]e^(-x^2)e^(-y^2)dxdy】
=√【∫[0,π】dθ∫[0.＋∞]e^(-ρ²)ρdρ
=√(π/2)