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

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

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

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

1 - \frac{1}{e}

Decimal

0.39346 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

\frac{1}{2} (- (- 2 [e^{u}]_{- \frac{1}{2}}^{0})) : [e^{u}]_{- \frac{1}{2}}^{0}

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

Compute the boundaries:

1 - \frac{1}{e}

= 1 - \frac{1}{e}

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