What is the integral from 0 to infinity of xe^{-x-y} ?

The integral from 0 to infinity of xe^{-x-y} is e^{-y}

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integral from 0 to infinity of xe^{-x-y}

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

e^{- y}

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- e^{- y})

Simplify

= e^{- y}

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What is the integral from 0 to infinity of xe^{-x-y} ?
The integral from 0 to infinity of xe^{-x-y} is e^{-y}