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

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

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

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

π erfi (\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} erfi (u)

= 2 \cdot \frac{π}{2} erfi (u)

Substitute back

u = \frac{x}{2}

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

Simplify

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

= π erfi (\frac{x}{2})

Add a constant to the solution

= π erfi (\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 sqrt(pi)\erfi(x/2)+C