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1. \int_{-\infty}^{\infty} \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. \int \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx = \frac{\sqrt{2}}{2 \sqrt{\pi}} \int e^{- \frac{x^{2}}{2}}\, dx

   Pull the constant \frac{\sqrt{2}}{2 \sqrt{\pi}} out of the integral.

3. \int e^{- \frac{x^{2}}{2}}\, dx = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}

   Erf rule.

4. F(\infty) - F(-\infty) = \left(\frac{1}{2}\right) - \left(- \frac{1}{2}\right)

   Fundamental theorem of calculus: plug in the limits.

5. = 1

   Simplify.