What is the integral from 0.5 to 1 of 2e^{-2x} ?

The integral from 0.5 to 1 of 2e^{-2x} is 1/e-1/(e^2)

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integral from 0.5 to 1 of 2e^{-2x}

\int_{0.5}^{1} 2 e^{- 2 x} d x

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

Decimal

0.23254 \dots

Solution steps

\int_{0.5}^{1} 2 e^{- 2 x} d x

Take the constant out:

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

= 2 \cdot \int_{0.5}^{1} e^{- 2 x} d x

Apply u-substitution

= 2 \cdot \int_{- 1}^{- 2} - \frac{1}{2} e^{u} d u

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

= 2 (- \int_{- 2}^{- 1} - \frac{1}{2} e^{u} d u)

Take the constant out:

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

= 2 (- (- \frac{1}{2} \cdot \int_{- 2}^{- 1} e^{u} d u))

Use the common integral:

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

= 2 (- (- \frac{1}{2} [e^{u}]_{- 2}^{- 1}))

Simplify

2 (- (- \frac{1}{2} [e^{u}]_{- 2}^{- 1})) : [e^{u}]_{- 2}^{- 1}

= [e^{u}]_{- 2}^{- 1}

Compute the boundaries:

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

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

What is the integral from 0.5 to 1 of 2e^{-2x} ?
The integral from 0.5 to 1 of 2e^{-2x} is 1/e-1/(e^2)