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

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

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integral of e^{-x^2 1/2}

\int e^{- x^{2} \frac{1}{2}} d x

\frac{π}{2} erf (\frac{x}{2}) + C

Solution steps

\int e^{- x^{2} \frac{1}{2}} d x

Simplify

- x^{2} \frac{1}{2} : - \frac{x ^{2}}{2}

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

= \frac{π}{2} erf (\frac{x}{2})

Add a constant to the solution

= \frac{π}{2} erf (\frac{x}{2}) + C

\frac{d}{d x} (\frac{2 x + 6 x ^{2}}{( 1 + 6 x ) ^{2}})

y + x^{2} + y^{2} d x - x d y = 0

y^{^{''}} - 3 y^{^{'}} - 4 y = 0, y (0) = a, y^{^{'}} (0) = - 20

\frac{d}{d x} (x^{\frac{8}{3}} - 20 x^{\frac{2}{3}})

\frac{d y}{d x} = y^{2} sin (x^{2}), y (- 2) = \frac{1}{3}

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