What is the integral from 2 to 4 of 0.5e^{-|x|} ?

The integral from 2 to 4 of 0.5e^{-|x|} is 0.05850…

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integral from 2 to 4 of 0.5e^{-|x|}

\int_{2}^{4} 0.5 e^{- ∣ x ∣} d x

0.05850 \dots

Solution steps

\int_{2}^{4} 0.5 e^{- ∣ x ∣} d x

Eliminate Absolutes

= \int_{2}^{4} 0.5 e^{- x} d x

Take the constant out:

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

= 0.5 \cdot \int_{2}^{4} e^{- x} d x

Apply u-substitution

: \int_{- 2}^{- 4} - e^{u} d u

= 0.5 \cdot \int_{- 2}^{- 4} - e^{u} d u

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

= 0.5 (- \int_{- 4}^{- 2} - e^{u} d u)

Take the constant out:

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

= 0.5 (- (- \int_{- 4}^{- 2} e^{u} d u))

Use the common integral:

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

= 0.5 (- (- [e^{u}]_{- 4}^{- 2}))

Simplify

= 0.5 [e^{u}]_{- 4}^{- 2}

Compute the boundaries:

\frac{1}{e ^{2}} - \frac{1}{e ^{4}}

= 0.5 (\frac{1}{e ^{2}} - \frac{1}{e ^{4}})

Simplify

= 0.05850 \dots

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What is the integral from 2 to 4 of 0.5e^{-|x|} ?
The integral from 2 to 4 of 0.5e^{-|x|} is 0.05850…