Solve any maths problem

Step by step

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

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

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

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

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

   Rewrite the integral in terms of u.

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

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

5. \int 1\, d_u = u

   The integral of a constant c is c·x.

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

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

7. F(\infty) - F(0) = \left(0\right) - \left(- \frac{1}{2}\right)

   Fundamental theorem of calculus: plug in the limits.

8. = \frac{1}{2}

   Simplify.

Reveal the answer

\frac{1}{2}