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

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

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

   Erf rule.

3. F(\infty) - F(-\infty) = \left(\frac{\sqrt{\pi}}{2}\right) - \left(- \frac{\sqrt{\pi}}{2}\right)

   Fundamental theorem of calculus: plug in the limits.

4. = \sqrt{\pi} \approx 1.7725

   Simplify.