What is the integral from 1 to infinity of xe^{-ax} ?

The integral from 1 to infinity of xe^{-ax} is -(-ae^{-a}-e^{-a})/(a^2)

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integral from 1 to infinity of xe^{-ax}

\int_{1}^{\infty} x e^{- a x} d x

- \frac{- a e ^{- a} - e ^{- a}}{a ^{2}}

Solution steps

\int_{1}^{\infty} x e^{- a x} d x

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

= - \frac{- a e ^{- a} - e ^{- a}}{a ^{2}}

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What is the integral from 1 to infinity of xe^{-ax} ?
The integral from 1 to infinity of xe^{-ax} is -(-ae^{-a}-e^{-a})/(a^2)