What is the integral from 0 to 6 of 4x^2e^{-x} ?

The integral from 0 to 6 of 4x^2e^{-x} is 4(-(50)/(e^6)+2)

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integral from 0 to 6 of 4x^2e^{-x}

\int_{0}^{6} 4 x^{2} e^{- x} d x

4 (- \frac{50}{e ^{6}} + 2)

Decimal

7.50424 \dots

Solution steps

\int_{0}^{6} 4 x^{2} e^{- x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{0}^{6} x^{2} e^{- x} d x

Apply Integration By Parts

= 4 [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{6}

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

= 4 [- e^{- x} x^{2} - (- 2 (- e^{- x} x - e^{- x}))]_{0}^{6}

Simplify

= 4 [- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})]_{0}^{6}

Compute the boundaries:

- \frac{50}{e ^{6}} + 2

= 4 (- \frac{50}{e ^{6}} + 2)

What is the integral from 0 to 6 of 4x^2e^{-x} ?
The integral from 0 to 6 of 4x^2e^{-x} is 4(-(50)/(e^6)+2)