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

The integral of-e^{-(x^2)/2} is -(pi/2)^{1/2}\erf(x/(sqrt(2)))+C

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integral of-e^{-(x^2)/2}

\int - e^{- \frac{x ^{2}}{2}} d x

- (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2}) + C

Solution steps

\int - 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

= - \int 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 e^{- (\frac{x}{2})^{2}} d x

Apply u-substitution

: \int e^{- u^{2}} 2 d u

= - \int 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 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)

Substitute back

u = \frac{x}{2}

= - 2 \frac{π}{2} erf (\frac{x}{2})

Simplify

- 2 \frac{π}{2} erf (\frac{x}{2}) : - (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2})

= - (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2})

Add a constant to the solution

= - (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2}) + C

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What is the integral of-e^{-(x^2)/2} ?
The integral of-e^{-(x^2)/2} is -(pi/2)^{1/2}\erf(x/(sqrt(2)))+C