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

The integral from x to infinity of ye^{x-y} is -e^x(-e^{-x}x-e^{-x})

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integral from x to infinity of ye^{x-y}

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

- e^{x} (- e^{- x} x - e^{- x})

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

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

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What is the integral from x to infinity of ye^{x-y} ?
The integral from x to infinity of ye^{x-y} is -e^x(-e^{-x}x-e^{-x})