What is the integral from 0 to 5 of 500xe^{-0.01x} ?

The integral from 0 to 5 of 500xe^{-0.01x} is 6045.52137…

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integral from 0 to 5 of 500xe^{-0.01x}

\int_{0}^{5} 500 x e^{- 0.01 x} d x

6045.52137 \dots

Solution steps

\int_{0}^{5} 500 x e^{- 0.01 x} d x

Take the constant out:

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

= 500 \cdot \int_{0}^{5} x e^{- 0.01 x} d x

Apply u-substitution

= 500 \cdot \int_{0}^{- 0.05} \frac{100}{0.01} e^{u} u d u

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

= 500 (- \int_{- 0.05}^{0} \frac{100}{0.01} e^{u} u d u)

Take the constant out:

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

= 500 (- \frac{100}{0.01} \cdot \int_{- 0.05}^{0} e^{u} u d u)

Simplify

= 500 (- 10000 \cdot \int_{- 0.05}^{0} e^{u} u d u)

Apply Integration By Parts

= 500 (- 10000 [e^{u} u - \int e^{u} d u]_{- 0.05}^{0})

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

= 500 (- 10000 [e^{u} u - e^{u}]_{- 0.05}^{0})

Simplify

= - 5000000 [e^{u} u - e^{u}]_{- 0.05}^{0}

Compute the boundaries:

- 0.00120 \dots

= - 5000000 (- 0.00120 \dots)

Simplify

= 6045.52137 \dots

What is the integral from 0 to 5 of 500xe^{-0.01x} ?
The integral from 0 to 5 of 500xe^{-0.01x} is 6045.52137…