What is the integral from 0 to 3 of (x-3)e^{-x} ?

The integral from 0 to 3 of (x-3)e^{-x} is -1/(e^3)-2

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integral from 0 to 3 of (x-3)e^{-x}

\int_{0}^{3} (x - 3) e^{- x} d x

- \frac{1}{e ^{3}} - 2

Decimal Notation

- 2.04978 \dots

Solution steps

\int_{0}^{3} (x - 3) e^{- x} d x

Apply Integration By Parts

: [- e^{- x} (x - 3) - \int - e^{- x} d x]_{0}^{3}

= [- e^{- x} (x - 3) - \int - e^{- x} d x]_{0}^{3}

\int - e^{- x} d x = e^{- x}

= [- e^{- x} (x - 3) - e^{- x}]_{0}^{3}

Expand

- e^{- x} (x - 3) - e^{- x} : - e^{- x} x + 2 e^{- x}

= [- e^{- x} x + 2 e^{- x}]_{0}^{3}

Compute the boundaries:

- \frac{1}{e ^{3}} - 2

= - \frac{1}{e ^{3}} - 2

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\int_{1}^{1} 1

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\int_{0}^{1} x^{2} (x + \frac{1}{2}) d x

What is the integral from 0 to 3 of (x-3)e^{-x} ?
The integral from 0 to 3 of (x-3)e^{-x} is -1/(e^3)-2