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

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

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

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

\frac{π}{2} erfi (\frac{x}{2}) + C

Solution steps

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

Simplify

\frac{1}{2} x^{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} erfi (u)

= 2 \frac{π}{2} erfi (u)

Substitute back

u = \frac{x}{2}

= 2 \frac{π}{2} erfi (\frac{x}{2})

Simplify

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

= \frac{π}{2} erfi (\frac{x}{2})

Add a constant to the solution

= \frac{π}{2} erfi (\frac{x}{2}) + C

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