Integral of 2*x*e^(-x^2) dx

1. The integral of a constant times a function is the constant times the integral of the function:

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

   1. Let $u = e^{- x^{2}}$.

      Then let $du = - 2 x e^{- x^{2}} dx$ and substitute $- \frac{du}{2}$:

      $$\int \frac{1}{4}\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 1\, du = u$$

         So, the result is: $- \frac{u}{2}$

      Now substitute $u$ back in:

      $$- \frac{e^{- x^{2}}}{2}$$

   So, the result is: $- e^{- x^{2}}$

2. Add the constant of integration:

   $$- e^{- x^{2}}+ \mathrm{constant}$$

The answer is:

$$- e^{- x^{2}}+ \mathrm{constant}$$