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

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

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

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

π erf (\frac{x}{2}) + C

Solution steps

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

Apply u-substitution

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

2 \cdot \frac{π}{2} erf (\frac{x}{2}) : π erf (\frac{x}{2})

= π erf (\frac{x}{2})

Add a constant to the solution

= π erf (\frac{x}{2}) + C

\frac{\partial}{\partial y} (3 x^{2} + y^{2})

y = (2 x + 1)^{2}

\frac{d ^{3}}{d x ^{3}} (cos (3 x))

\int_{1}^{\infty} \frac{1}{2 ^{x}} d x

(2 x^{2} + 1)^{3} (x^{2} - 4)^{2}

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