Integral of 2xe^(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:

         $$\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$$

         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:

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