Integral of 2x-2x^2 dx

1. Integrate term-by-term:

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

      $$\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

      So, the result is: $- \frac{2 x^{3}}{3}$

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

      $$\int 2 x\, dx = 2 \int x\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x\, dx = \frac{x^{2}}{2}$$

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

   The result is: $- \frac{2 x^{3}}{3} + x^{2}$

2. Now simplify:

   $$\frac{x^{2} \left(3 - 2 x\right)}{3}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(3 - 2 x\right)}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(3 - 2 x\right)}{3}+ \mathrm{constant}$$