integral from 0 to infinity of 4xye^{-(x^2+y^2)}

解答

\int_{0}^{\infty} 4 x y e^{- (x^{2} + y^{2})} d y

2 e^{- x^{2}} x

求解步骤

\int_{0}^{\infty} 4 x y e^{- (x^{2} + y^{2})} d y

计算不定积分:

\int 4 x y e^{- (x^{2} + y^{2})} d y = - 2 x e^{- x^{2} - y^{2}} + C

带入上下限计算后得:

\int_{0}^{\infty} 4 x y e^{- (x^{2} + y^{2})} d y = 0 - (- 2 e^{- x^{2}} x)

= 0 - (- 2 e^{- x^{2}} x)

化简

= 2 e^{- x^{2}} x

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