What is the integral from 1 to infinity of x^2e^{-x} ?

The integral from 1 to infinity of x^2e^{-x} is 5/e

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integral from 1 to infinity of x^2e^{-x}

\int_{1}^{\infty} x^{2} e^{- x} d x

\frac{5}{e}

Decimal

1.83939 \dots

Solution steps

\int_{1}^{\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_{1}^{\infty} x^{2} e^{- x} d x = 0 - (- \frac{5}{e})

= 0 - (- \frac{5}{e})

Simplify

= \frac{5}{e}

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What is the integral from 1 to infinity of x^2e^{-x} ?
The integral from 1 to infinity of x^2e^{-x} is 5/e