Integral of 2*x^3-0.5*x dx

1. Integrate term-by-term:

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

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

      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: $\frac{x^{4}}{2}$

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

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

      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: $- \frac{x^{2}}{4}$

   The result is: $\frac{x^{4}}{2} - \frac{x^{2}}{4}$

2. Add the constant of integration:

   $$\frac{x^{4}}{2} - \frac{x^{2}}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{4}}{2} - \frac{x^{2}}{4}+ \mathrm{constant}$$