What is the integral from 0 to 12 of 8xe^{-0.2x} ?

The integral from 0 to 12 of 8xe^{-0.2x} is 138.31179…

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integral from 0 to 12 of 8xe^{-0.2x}

\int_{0}^{12} 8 x e^{- 0.2 x} d x

138.31179 \dots

Solution steps

\int_{0}^{12} 8 x e^{- 0.2 x} d x

Take the constant out:

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

= 8 \cdot \int_{0}^{12} x e^{- 0.2 x} d x

Apply u-substitution

: \int_{0}^{- 2.4} \frac{5}{0.2} e^{u} u d u

= 8 \cdot \int_{0}^{- 2.4} \frac{5}{0.2} e^{u} u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 8 (- \int_{- 2.4}^{0} \frac{5}{0.2} e^{u} u d u)

Take the constant out:

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

= 8 (- \frac{5}{0.2} \cdot \int_{- 2.4}^{0} e^{u} u d u)

Simplify

= 8 (- 25 \cdot \int_{- 2.4}^{0} e^{u} u d u)

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{- 2.4}^{0}

= 8 (- 25 [e^{u} u - \int e^{u} d u]_{- 2.4}^{0})

\int e^{u} d u = e^{u}

= 8 (- 25 [e^{u} u - e^{u}]_{- 2.4}^{0})

Simplify

= - 200 [e^{u} u - e^{u}]_{- 2.4}^{0}

Compute the boundaries:

- 0.69155 \dots

= - 200 (- 0.69155 \dots)

Simplify

= 138.31179 \dots

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What is the integral from 0 to 12 of 8xe^{-0.2x} ?
The integral from 0 to 12 of 8xe^{-0.2x} is 138.31179…