What is the integral from 0 to 4 of xe^{-0.1x} ?

The integral from 0 to 4 of xe^{-0.1x} is 6.15519…

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integral from 0 to 4 of xe^{-0.1x}

\int_{0}^{4} x e^{- 0.1 x} d x

6.15519 \dots

Solution steps

\int_{0}^{4} x e^{- 0.1 x} d x

Apply u-substitution

: \int_{0}^{- 0.4} \frac{10}{0.1} e^{u} u d u

= \int_{0}^{- 0.4} \frac{10}{0.1} e^{u} u d u

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

= - \int_{- 0.4}^{0} \frac{10}{0.1} e^{u} u d u

Take the constant out:

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

= - \frac{10}{0.1} \cdot \int_{- 0.4}^{0} e^{u} u d u

Simplify

= - 100 \cdot \int_{- 0.4}^{0} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{- 0.4}^{0}

= - 100 [e^{u} u - \int e^{u} d u]_{- 0.4}^{0}

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

= - 100 [e^{u} u - e^{u}]_{- 0.4}^{0}

Compute the boundaries:

- 0.06155 \dots

= - 100 (- 0.06155 \dots)

Simplify

= 6.15519 \dots

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What is the integral from 0 to 4 of xe^{-0.1x} ?
The integral from 0 to 4 of xe^{-0.1x} is 6.15519…