What is the integral from 0 to infinity of λxe^{-λx} ?

The integral from 0 to infinity of λxe^{-λx} is 1/λ

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integral from 0 to infinity of λxe^{-λx}

\int_{0}^{\infty} λ x e^{- λ x} d x

\frac{1}{λ}

Solution steps

\int_{0}^{\infty} λ x e^{- λ x} d x

Compute the indefinite integral:

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

Compute the boundaries:

\int_{0}^{\infty} λ x e^{- λ x} d x = 0 - (- \frac{1}{λ})

= 0 - (- \frac{1}{λ})

Simplify

= \frac{1}{λ}

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What is the integral from 0 to infinity of λxe^{-λx} ?
The integral from 0 to infinity of λxe^{-λx} is 1/λ