What is the integral from a to infinity of x^2e^{-x} ?

The integral from a to infinity of x^2e^{-x} is 2ae^{-a}+2e^{-a}+a^2e^{-a}

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integral from a to infinity of x^2e^{-x}

\int_{a}^{\infty} x^{2} e^{- x} d x

2 a e^{- a} + 2 e^{- a} + a^{2} e^{- a}

Solution steps

\int_{a}^{\infty} x^{2} e^{- x} d x

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- e^{- a} a^{2} + 2 (- e^{- a} a - e^{- a}))

Simplify

= 2 a e^{- a} + 2 e^{- a} + a^{2} e^{- a}

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What is the integral from a to infinity of x^2e^{-x} ?
The integral from a to infinity of x^2e^{-x} is 2ae^{-a}+2e^{-a}+a^2e^{-a}