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

The integral from-1 to 1 of 1/(sqrt(2pi))e^{-(x^2)/2} is 0.68268…

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

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

0.68268 \dots

Solution steps

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

Take the constant out:

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

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

2 π = 2^{\frac{1}{2}} π^{\frac{1}{2}}

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} \cdot \int_{- 1}^{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}

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

Apply u-substitution

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

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} \cdot \int_{- \frac{1}{2}}^{\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

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

Use the nonelementary integral:

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

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

Simplify

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

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

Compute the boundaries:

1.21003 \dots

= \frac{1}{π} \cdot 1.21003 \dots

Simplify

= 0.68268 \dots

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What is the integral from-1 to 1 of 1/(sqrt(2pi))e^{-(x^2)/2} ?
The integral from-1 to 1 of 1/(sqrt(2pi))e^{-(x^2)/2} is 0.68268…