What is the integral from 0.5 to 1 of 3e^{-3x} ?

The integral from 0.5 to 1 of 3e^{-3x} is 0.17334…

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integral from 0.5 to 1 of 3e^{-3x}

\int_{0.5}^{1} 3 e^{- 3 x} d x

0.17334 \dots

Solution steps

\int_{0.5}^{1} 3 e^{- 3 x} d x

Take the constant out:

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

= 3 \cdot \int_{0.5}^{1} e^{- 3 x} d x

Apply u-substitution

: \int_{- 1.5}^{- 3} - \frac{1}{3} e^{u} d u

= 3 \cdot \int_{- 1.5}^{- 3} - \frac{1}{3} e^{u} d u

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

= 3 (- \int_{- 3}^{- 1.5} - \frac{1}{3} e^{u} d u)

Take the constant out:

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

= 3 (- (- \frac{1}{3} \cdot \int_{- 3}^{- 1.5} e^{u} d u))

Use the common integral:

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

= 3 (- (- \frac{1}{3} [e^{u}]_{- 3}^{- 1.5}))

Simplify

3 (- (- \frac{1}{3} [e^{u}]_{- 3}^{- 1.5})) : [e^{u}]_{- 3}^{- 1.5}

= [e^{u}]_{- 3}^{- 1.5}

Compute the boundaries:

0.17334 \dots

= 0.17334 \dots

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What is the integral from 0.5 to 1 of 3e^{-3x} ?
The integral from 0.5 to 1 of 3e^{-3x} is 0.17334…