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积分上下限为0到正无穷的x^m*exp[-x^2]dx 的定积分:
(1)当m是奇数,∫x^m*exp[-x^2]dx =1/2 *∫x^(m-1)*exp[-x^2]dx^2 =1/2*[(m-1)/2]!
(2)当m是偶数,∫x^m*exp[-x^2]dx =√∫x^m*exp[-x^2]dx ∫y^m*exp[-y^2]dy=√∫(sinθcosθ)^mdθ∫r^2m*exp[-r^2]rdr=√(1/2^m)*∫(sin2θ)^mdθ * (m!/2)(dθ积分上下限为0到π/2)=√(1/2^m)*(m-1)!/m!*(π/2)*(m!/2)=(m-1)!* √π /2^[m/2+1]
1年前
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