What is the integral from 6 to 11 of 10xe^{-0.2x} ?

The integral from 6 to 11 of 10xe^{-0.2x} is 77.01428…

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integral from 6 to 11 of 10xe^{-0.2x}

\int_{6}^{11} 10 x e^{- 0.2 x} d x

77.01428 \dots

Solution steps

\int_{6}^{11} 10 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

= 10 \cdot \int_{6}^{11} x e^{- 0.2 x} d x

Apply u-substitution

= 10 \cdot \int_{- 1.2}^{- 2.2} \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

= 10 (- \int_{- 2.2}^{- 1.2} \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

= 10 (- \frac{5}{0.2} \cdot \int_{- 2.2}^{- 1.2} e^{u} u d u)

Simplify

= 10 (- 25 \cdot \int_{- 2.2}^{- 1.2} e^{u} u d u)

Apply Integration By Parts

= 10 (- 25 [e^{u} u - \int e^{u} d u]_{- 2.2}^{- 1.2})

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

= 10 (- 25 [e^{u} u - e^{u}]_{- 2.2}^{- 1.2})

Simplify

= - 250 [e^{u} u - e^{u}]_{- 2.2}^{- 1.2}

Compute the boundaries:

- 0.30805 \dots

= - 250 (- 0.30805 \dots)

Simplify

= 77.01428 \dots

What is the integral from 6 to 11 of 10xe^{-0.2x} ?
The integral from 6 to 11 of 10xe^{-0.2x} is 77.01428…