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

The integral from-1/2 to 1/2 of x^2e^{-x} is -(13}{4sqrt(e)}+\frac{sqrt(e))/4+sqrt(e)

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

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

- \frac{13}{4 e} + \frac{e}{4} + e

Decimal

0.08967 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Compute the boundaries:

- \frac{13}{4 e} + \frac{e}{4} + e

= - \frac{13}{4 e} + \frac{e}{4} + e

What is the integral from-1/2 to 1/2 of x^2e^{-x} ?
The integral from-1/2 to 1/2 of x^2e^{-x} is -(13}{4sqrt(e)}+\frac{sqrt(e))/4+sqrt(e)