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Calculus · Definite-Integrals

Integrate sqrt(2)·e^(-x^2/2)/(2·sqrt(pi)) from 0 to 1

\int_{0}^{1} \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx

Krok za krokem

  1. \int_{0}^{1} \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(1) - F(0) = \left(\frac{\operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{2}\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  5. = \frac{\operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \approx 0.34134

    Simplify.

Odhalte odpověď
\frac{\operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{2}