What is the integral from θ to infinity of xe^{-x+θ} ?

The integral from θ to infinity of xe^{-x+θ} is -e^θ(-e^{-θ}θ-e^{-θ})

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integral from θ to infinity of xe^{-x+θ}

\int_{θ}^{\infty} x e^{- x + θ} d x

- e^{θ} (- e^{- θ} θ - e^{- θ})

Solution steps

\int_{θ}^{\infty} x e^{- x + θ} d x

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - e^{θ} (- e^{- θ} θ - e^{- θ})

Simplify

= - e^{θ} (- e^{- θ} θ - e^{- θ})

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What is the integral from θ to infinity of xe^{-x+θ} ?
The integral from θ to infinity of xe^{-x+θ} is -e^θ(-e^{-θ}θ-e^{-θ})