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

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

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

\int_{θ}^{\infty} x e^{- x} d x

θ e^{- θ} + e^{- θ}

Solution steps

\int_{θ}^{\infty} x e^{- x} d x

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- e^{- θ} θ - e^{- θ})

Simplify

= θ e^{- θ} + e^{- θ}

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