What is the integral from 0 to infinity of xye^{-xy} ?

The integral from 0 to infinity of xye^{-xy} is 1/x

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

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

\frac{1}{x}

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

= \frac{1}{x}

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What is the integral from 0 to infinity of xye^{-xy} ?
The integral from 0 to infinity of xye^{-xy} is 1/x