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

1. Rewrite the integrand:

   $$x^{2} \left(x - 2\right)^{2} = x^{4} - 4 x^{3} + 4 x^{2}$$

2. Integrate term-by-term:

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

      $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

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

      $$\int \left(- 4 x^{3}\right)\, dx = - 4 \int x^{3}\, dx$$

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

         $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

      So, the result is: $- x^{4}$

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

      $$\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$$

      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: $\frac{4 x^{3}}{3}$

   The result is: $\frac{x^{5}}{5} - x^{4} + \frac{4 x^{3}}{3}$

3. Now simplify:

   $$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)$$

4. Add the constant of integration:

   $$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)+ \mathrm{constant}$$

The answer is:

$$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)+ \mathrm{constant}$$