Question

求这道题的步骤！

求这道题的步骤！求这道题的步骤！

Answer 1

如图示

Answer 2

令：F = (-∞,+∞)∫e^(-x²)dx
同样 F= (-∞,+∞)∫e^(-y²)dy
由于x,y是互不相关的的积分变量,因此：
F² = (-∞,+∞)∫e^(-x²)dx * (-∞,+∞)∫e^(-y²)dy
= [D]∫∫e^(-x²)*dx * e^(-y²)*dy
= [D]∫∫e^[-(x²+y²)]*dx *dy
式中积分域D = {(x,y)|x ∈(-∞,+∞),y∈(-∞,+∞)}
对x,y进行极坐标变换,则:
x²+y² = ρ²；dxdy = ρ*dρ*dθ
F² = [D]∫∫e^[-(x²+y²)]*dx *dy
= [0,+∞)[0,2π]∫∫e^(-ρ²) ρ*dρ*dθ
= [0,2π]∫dθ *(0,+∞)∫e^(-ρ²) ρ*dρ
= 2π* 1/2*[0,+∞)*∫e^(-ρ²) *dρ²
= π
因此 F = (-∞,+∞)∫e^(-x²)dx = √π
若满意，请采纳。

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