integral from 0 to infinity of xye^{-xy}

解答

\int_{0}^{\infty} x y e^{- x y} d y

\frac{1}{x}

求解步骤

\int_{0}^{\infty} x y e^{- x y} d y

计算不定积分:

\int x y e^{- x y} d y = - e^{- x y} y - \frac{e ^{- x y}}{x} + C

带入上下限计算后得:

\int_{0}^{\infty} x y e^{- x y} d y = 0 - (- \frac{1}{x})

= 0 - (- \frac{1}{x})

化简

= \frac{1}{x}

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