What is the integral from 0 to 5 of x^2e^{-x} ?

The integral from 0 to 5 of x^2e^{-x} is -(37)/(e^5)+2

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integral from 0 to 5 of x^2e^{-x}

\int_{0}^{5} x^{2} e^{- x} d x

- \frac{37}{e ^{5}} + 2

Decimal

1.75069 \dots

Solution steps

\int_{0}^{5} x^{2} e^{- x} d x

Apply Integration By Parts

: [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{5}

= [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{5}

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

= [- e^{- x} x^{2} - (- 2 (- e^{- x} x - e^{- x}))]_{0}^{5}

Simplify

= [- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})]_{0}^{5}

Compute the boundaries:

- \frac{37}{e ^{5}} + 2

= - \frac{37}{e ^{5}} + 2

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What is the integral from 0 to 5 of x^2e^{-x} ?
The integral from 0 to 5 of x^2e^{-x} is -(37)/(e^5)+2