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Calculus · Definite-Integrals

Integrate e^(-2x^2) from -oo to oo

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

Step by step

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

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

  2. \int e^{- 2 x^{2}}\, dx = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\sqrt{2} x \right)}}{4}

    Erf rule.

  3. F(\infty) - F(-\infty) = \left(\frac{\sqrt{2} \sqrt{\pi}}{4}\right) - \left(- \frac{\sqrt{2} \sqrt{\pi}}{4}\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{\sqrt{2} \sqrt{\pi}}{2} \approx 1.2533

    Simplify.

Reveal the answer
\frac{\sqrt{2} \sqrt{\pi}}{2}