What is the integral from 100 to 200 of 0.1e^{-0.002x} ?

The integral from 100 to 200 of 0.1e^{-0.002x} is 7.42053…

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integral from 100 to 200 of 0.1e^{-0.002x}

\int_{100}^{200} 0.1 e^{- 0.002 x} d x

7.42053 \dots

Solution steps

\int_{100}^{200} 0.1 e^{- 0.002 x} d x

Take the constant out:

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

= 0.1 \cdot \int_{100}^{200} e^{- 0.002 x} d x

Apply u-substitution

: \int_{- 0.2}^{- 0.4} - \frac{1}{0.002} e^{u} d u

= 0.1 \cdot \int_{- 0.2}^{- 0.4} - \frac{1}{0.002} e^{u} d u

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

= 0.1 (- \int_{- 0.4}^{- 0.2} - \frac{1}{0.002} e^{u} d u)

Take the constant out:

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

= 0.1 (- (- \frac{1}{0.002} \cdot \int_{- 0.4}^{- 0.2} e^{u} d u))

Simplify

= 0.1 (- (- 500 \cdot \int_{- 0.4}^{- 0.2} e^{u} d u))

Use the common integral:

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

= 0.1 (- (- 500 [e^{u}]_{- 0.4}^{- 0.2}))

Simplify

= 50 [e^{u}]_{- 0.4}^{- 0.2}

Compute the boundaries:

0.14841 \dots

= 50 \cdot 0.14841 \dots

Simplify

= 7.42053 \dots

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What is the integral from 100 to 200 of 0.1e^{-0.002x} ?
The integral from 100 to 200 of 0.1e^{-0.002x} is 7.42053…