What is the integral from 0 to 15 of 1000e^{-0.05x} ?

The integral from 0 to 15 of 1000e^{-0.05x} is 10552.66894…

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integral from 0 to 15 of 1000e^{-0.05x}

\int_{0}^{15} 1000 e^{- 0.05 x} d x

10552.66894 \dots

Solution steps

\int_{0}^{15} 1000 e^{- 0.05 x} d x

Take the constant out:

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

= 1000 \cdot \int_{0}^{15} e^{- 0.05 x} d x

Apply u-substitution

: \int_{0}^{- 0.75} - \frac{1}{0.05} e^{u} d u

= 1000 \cdot \int_{0}^{- 0.75} - \frac{1}{0.05} e^{u} d u

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

= 1000 (- \int_{- 0.75}^{0} - \frac{1}{0.05} e^{u} d u)

Take the constant out:

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

= 1000 (- (- \frac{1}{0.05} \cdot \int_{- 0.75}^{0} e^{u} d u))

Simplify

= 1000 (- (- 20 \cdot \int_{- 0.75}^{0} e^{u} d u))

Use the common integral:

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

= 1000 (- (- 20 [e^{u}]_{- 0.75}^{0}))

Simplify

= 20000 [e^{u}]_{- 0.75}^{0}

Compute the boundaries:

0.52763 \dots

= 20000 \cdot 0.52763 \dots

Simplify

= 10552.66894 \dots

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What is the integral from 0 to 15 of 1000e^{-0.05x} ?
The integral from 0 to 15 of 1000e^{-0.05x} is 10552.66894…