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

The integral from 0 to infinity of xe^{-x/θ} is θ^2

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

\int_{0}^{\infty} x e^{- \frac{x}{θ}} d x

θ^{2}

Solution steps

\int_{0}^{\infty} x e^{- \frac{x}{θ}} d x

Compute the indefinite integral:

\int x e^{- \frac{x}{θ}} d x = - θ e^{- \frac{x}{θ}} x - θ^{2} e^{- \frac{x}{θ}} + C

Compute the boundaries:

\int_{0}^{\infty} x e^{- \frac{x}{θ}} d x = 0 - (- θ^{2})

= 0 - (- θ^{2})

Simplify

= θ^{2}

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What is the integral from 0 to infinity of xe^{-x/θ} ?
The integral from 0 to infinity of xe^{-x/θ} is θ^2