What is the integral from 7 to 14 of 0.05e^{-0.05x} ?

The integral from 7 to 14 of 0.05e^{-0.05x} is 0.20810…

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integral from 7 to 14 of 0.05e^{-0.05x}

\int_{7}^{14} 0.05 e^{- 0.05 x} d x

0.20810 \dots

Solution steps

\int_{7}^{14} 0.05 e^{- 0.05 x} d x

Take the constant out:

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

= 0.05 \cdot \int_{7}^{14} e^{- 0.05 x} d x

Apply u-substitution

: \int_{- 0.35}^{- 0.7} - \frac{1}{0.05} e^{u} d u

= 0.05 \cdot \int_{- 0.35}^{- 0.7} - \frac{1}{0.05} e^{u} d u

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

= 0.05 (- \int_{- 0.7}^{- 0.35} - \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

= 0.05 (- (- \frac{1}{0.05} \cdot \int_{- 0.7}^{- 0.35} e^{u} d u))

Simplify

= 0.05 (- (- 20 \cdot \int_{- 0.7}^{- 0.35} e^{u} d u))

Use the common integral:

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

= 0.05 (- (- 20 [e^{u}]_{- 0.7}^{- 0.35}))

Simplify

= [e^{u}]_{- 0.7}^{- 0.35}

Compute the boundaries:

0.20810 \dots

= 0.20810 \dots

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What is the integral from 7 to 14 of 0.05e^{-0.05x} ?
The integral from 7 to 14 of 0.05e^{-0.05x} is 0.20810…