What is the integral from 0 to 5 of 250000e^{-0.1x} ?

The integral from 0 to 5 of 250000e^{-0.1x} is 983673.35071…

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integral from 0 to 5 of 250000e^{-0.1x}

\int_{0}^{5} 250000 e^{- 0.1 x} d x

983673.35071 \dots

Solution steps

\int_{0}^{5} 250000 e^{- 0.1 x} d x

Take the constant out:

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

= 250000 \cdot \int_{0}^{5} e^{- 0.1 x} d x

Apply u-substitution

: \int_{0}^{- 0.5} - \frac{1}{0.1} e^{u} d u

= 250000 \cdot \int_{0}^{- 0.5} - \frac{1}{0.1} e^{u} d u

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

= 250000 (- \int_{- 0.5}^{0} - \frac{1}{0.1} e^{u} d u)

Take the constant out:

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

= 250000 (- (- \frac{1}{0.1} \cdot \int_{- 0.5}^{0} e^{u} d u))

Simplify

= 250000 (- (- 10 \cdot \int_{- 0.5}^{0} e^{u} d u))

Use the common integral:

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

= 250000 (- (- 10 [e^{u}]_{- 0.5}^{0}))

Simplify

= 2500000 [e^{u}]_{- 0.5}^{0}

Compute the boundaries:

0.39346 \dots

= 2500000 \cdot 0.39346 \dots

Simplify

= 983673.35071 \dots

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What is the integral from 0 to 5 of 250000e^{-0.1x} ?
The integral from 0 to 5 of 250000e^{-0.1x} is 983673.35071…