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

The integral from 0 to 1 of e^{-(x^2)/2} is 0.85562…

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

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

0.85562 \dots

Solution steps

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

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the nonelementary integral:

\int e^{- u^{2}} d u = \frac{π}{2} erf (u)

= 2 [\frac{π}{2} erf (u)]_{0}^{\frac{1}{2}}

Compute the boundaries:

0.60501 \dots

= 2 \cdot 0.60501 \dots

Simplify

= 0.85562 \dots

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What is the integral from 0 to 1 of e^{-(x^2)/2} ?
The integral from 0 to 1 of e^{-(x^2)/2} is 0.85562…