What is the integral from 0 to 1 of e^{-((x-0.5))/2} ?

The integral from 0 to 1 of e^{-((x-0.5))/2} is 1.01044…

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to 1 of e^{-((x-0.5))/2}

\int_{0}^{1} e^{- \frac{( x - 0.5 )}{2}} d x

1.01044 \dots

Solution steps

\int_{0}^{1} e^{- \frac{x - 0.5}{2}} d x

- \frac{x - 0.5}{2} = \frac{1}{2} (- x - 1.5) + 1

= \int_{0}^{1} e^{\frac{1}{2} (- x - 1.5) + 1} d x

Simplify

e^{\frac{1}{2} (- x - 1.5) + 1} : e^{- 0.5 x + 0.25}

= \int_{0}^{1} e^{- 0.5 x + 0.25} d x

Apply u-substitution

= \int_{0.25}^{- 0.25} - \frac{1}{0.5} e^{u} d u

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

= - \int_{- 0.25}^{0.25} - \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

= - (- \frac{1}{0.5} \cdot \int_{- 0.25}^{0.25} e^{u} d u)

Simplify

= - (- 2 \cdot \int_{- 0.25}^{0.25} e^{u} d u)

Use the common integral:

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

= - (- 2 [e^{u}]_{- 0.25}^{0.25})

Simplify

= 2 [e^{u}]_{- 0.25}^{0.25}

Compute the boundaries:

0.50522 \dots

= 2 \cdot 0.50522 \dots

Simplify

= 1.01044 \dots

What is the integral from 0 to 1 of e^{-((x-0.5))/2} ?
The integral from 0 to 1 of e^{-((x-0.5))/2} is 1.01044…