What is the integral from 0 to infinity of xe^{-px} ?

The integral from 0 to infinity of xe^{-px} is 1/(p^2)

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integral from 0 to infinity of xe^{-px}

\int_{0}^{\infty} x e^{- p x} d x

\frac{1}{p ^{2}}

Solution steps

\int_{0}^{\infty} x e^{- p x} d x

Compute the indefinite integral:

\int x e^{- p x} d x = \frac{1}{p ^{2}} (- p e^{- p x} x - e^{- p x}) + C

Compute the boundaries:

\int_{0}^{\infty} x e^{- p x} d x = 0 - (- \frac{1}{p ^{2}})

= 0 - (- \frac{1}{p ^{2}})

Simplify

= \frac{1}{p ^{2}}

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What is the integral from 0 to infinity of xe^{-px} ?
The integral from 0 to infinity of xe^{-px} is 1/(p^2)