What is the integral from 0 to 1 of x^2*e^{-x} ?

The integral from 0 to 1 of x^2*e^{-x} is -5/e+2

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integral from 0 to 1 of x^2*e^{-x}

\int_{0}^{1} x^{2} \cdot e^{- x} d x

- \frac{5}{e} + 2

Decimal

0.16060 \dots

Solution steps

\int_{0}^{1} x^{2} e^{- x} d x

Apply Integration By Parts

: [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{1}

= [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{1}

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

= [- e^{- x} x^{2} - (- 2 (- e^{- x} x - e^{- x}))]_{0}^{1}

Simplify

= [- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})]_{0}^{1}

Compute the boundaries:

- \frac{5}{e} + 2

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

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What is the integral from 0 to 1 of x^2*e^{-x} ?
The integral from 0 to 1 of x^2*e^{-x} is -5/e+2