What is the integral from 0 to 52 of 3.75e^{-0.008x} ?

The integral from 0 to 52 of 3.75e^{-0.008x} is 159.52487…

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integral from 0 to 52 of 3.75e^{-0.008x}

\int_{0}^{52} 3.75 e^{- 0.008 x} d x

159.52487 \dots

Solution steps

\int_{0}^{52} 3.75 e^{- 0.008 x} d x

Take the constant out:

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

= 3.75 \cdot \int_{0}^{52} e^{- 0.008 x} d x

Apply u-substitution

: \int_{0}^{- 0.416} - \frac{1}{0.008} e^{u} d u

= 3.75 \cdot \int_{0}^{- 0.416} - \frac{1}{0.008} e^{u} d u

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

= 3.75 (- \int_{- 0.416}^{0} - \frac{1}{0.008} e^{u} d u)

Take the constant out:

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

= 3.75 (- (- \frac{1}{0.008} \cdot \int_{- 0.416}^{0} e^{u} d u))

Simplify

= 3.75 (- (- 125 \cdot \int_{- 0.416}^{0} e^{u} d u))

Use the common integral:

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

= 3.75 (- (- 125 [e^{u}]_{- 0.416}^{0}))

Simplify

= 468.75 [e^{u}]_{- 0.416}^{0}

Compute the boundaries:

0.34031 \dots

= 468.75 \cdot 0.34031 \dots

Simplify

= 159.52487 \dots

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What is the integral from 0 to 52 of 3.75e^{-0.008x} ?
The integral from 0 to 52 of 3.75e^{-0.008x} is 159.52487…