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Calculus · Definite-Integrals

Integrate 2x·e^(-x^2) from 0 to oo

\int_{0}^{\infty} 2 x e^{- x^{2}}\, dx

Step by step

  1. \int_{0}^{\infty} 2 x e^{- x^{2}}\, dx

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

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

    Pull the constant 2 out of the integral.

  3. u = e^{- x^{2}},\quad du = - 2 x e^{- x^{2}}\, dx

    Substitute u = e^{- x^{2}}.

  4. \int x e^{- x^{2}}\, dx = \int - \frac{1}{2}\, d_u

    Rewrite the integral in terms of u.

  5. \int - \frac{1}{2}\, d_u = - \frac{1}{2} \int 1\, d_u

    Pull the constant - \frac{1}{2} out of the integral.

  6. \int 1\, d_u = u

    The integral of a constant c is c·x.

  7. = - \frac{e^{- x^{2}}}{2}

    Substitute back u = e^{- x^{2}}.

  8. F(\infty) - F(0) = \left(0\right) - \left(-1\right)

    Fundamental theorem of calculus: plug in the limits.

  9. = 1

    Simplify.

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