What is the integral from 0 to 3 of 0.5e^{-0.5x} ?

The integral from 0 to 3 of 0.5e^{-0.5x} is 0.77686…

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integral from 0 to 3 of 0.5e^{-0.5x}

\int_{0}^{3} 0.5 e^{- 0.5 x} d x

0.77686 \dots

Solution steps

\int_{0}^{3} 0.5 e^{- 0.5 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_{0}^{3} e^{- 0.5 x} d x

Apply u-substitution

: \int_{0}^{- 1.5} - \frac{1}{0.5} e^{u} d u

= 0.5 \cdot \int_{0}^{- 1.5} - \frac{1}{0.5} e^{u} d u

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

= 0.5 (- \int_{- 1.5}^{0} - \frac{1}{0.5} e^{u} d u)

Take the constant out:

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

= 0.5 (- (- \frac{1}{0.5} \cdot \int_{- 1.5}^{0} e^{u} d u))

Simplify

= 0.5 (- (- 2 \cdot \int_{- 1.5}^{0} e^{u} d u))

Use the common integral:

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

= 0.5 (- (- 2 [e^{u}]_{- 1.5}^{0}))

Simplify

= [e^{u}]_{- 1.5}^{0}

Compute the boundaries:

0.77686 \dots

= 0.77686 \dots

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What is the integral from 0 to 3 of 0.5e^{-0.5x} ?
The integral from 0 to 3 of 0.5e^{-0.5x} is 0.77686…