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

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The integral $$ \int e^{-\frac{1}{2} x^2} \, dx $$ is a well-known integral that does not have a simple expression in terms of elementary functions. However, it can be expressed in terms of the error function, denoted as $$\text{erf}(x)$$.

The indefinite integral can be represented as:

$$
\int e^{-\frac{1}{2} x^2} \, dx = \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) + C
$$

where $$C$$ is the constant of integration.

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The integral $$ \int e^{-\frac{1}{2} x^2} \, dx $$ cannot be expressed using simple functions. It is typically written in terms of the error function as $$ \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) + C $$, where $$C$$ is a constant.

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