Integral of x(x-x^3) dx

1. Rewrite the integrand:

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

2. Integrate term-by-term:

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

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

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

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

      So, the result is: $- \frac{x^{5}}{5}$

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

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

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

3. Add the constant of integration:

   $$- \frac{x^{5}}{5} + \frac{x^{3}}{3}+ \mathrm{constant}$$

The answer is:

$$- \frac{x^{5}}{5} + \frac{x^{3}}{3}+ \mathrm{constant}$$