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Calculus · Definite-Integrals

Integrate sqrt(2)·e^(-x^2/2)/(2·sqrt(pi)) from -2 to 2

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

Step by step

  1. \int_{-2}^{2} \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(2) - F(-2) = \left(\frac{\operatorname{erf}{\left(\sqrt{2} \right)}}{2}\right) - \left(- \frac{\operatorname{erf}{\left(\sqrt{2} \right)}}{2}\right)

    Fundamental theorem of calculus: plug in the limits.

  5. = \operatorname{erf}{\left(\sqrt{2} \right)} \approx 0.95450

    Simplify.

Reveal the answer
\operatorname{erf}{\left(\sqrt{2} \right)}