What is the integral from 0 to 1 of 2e^{-x} ?

The integral from 0 to 1 of 2e^{-x} is 2(1-1/e)

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integral from 0 to 1 of 2e^{-x}

\int_{0}^{1} 2 e^{- x} d x

2 (1 - \frac{1}{e})

Decimal

1.26424 \dots

Solution steps

\int_{0}^{1} 2 e^{- 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}^{1} e^{- x} d x

Apply u-substitution

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

= 2 \cdot \int_{0}^{- 1} - e^{u} d u

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

= 2 (- \int_{- 1}^{0} - e^{u} d u)

Take the constant out:

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

= 2 (- (- \int_{- 1}^{0} e^{u} d u))

Use the common integral:

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

= 2 (- (- [e^{u}]_{- 1}^{0}))

Simplify

= 2 [e^{u}]_{- 1}^{0}

Compute the boundaries:

1 - \frac{1}{e}

= 2 (1 - \frac{1}{e})

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What is the integral from 0 to 1 of 2e^{-x} ?
The integral from 0 to 1 of 2e^{-x} is 2(1-1/e)