What is the integral from 0 to infinity of xθe^{-θx} ?

The integral from 0 to infinity of xθe^{-θx} is 1/θ

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integral from 0 to infinity of xθe^{-θx}

\int_{0}^{\infty} x θ e^{- θ x} d x

\frac{1}{θ}

Solution steps

\int_{0}^{\infty} x θ e^{- θ x} d x

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- \frac{1}{θ})

Simplify

= \frac{1}{θ}

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\int_{0}^{3} \frac{1}{9} x^{2} d x

What is the integral from 0 to infinity of xθe^{-θx} ?
The integral from 0 to infinity of xθe^{-θx} is 1/θ