Integral of (x-3)/2 dx

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

   $$\int \frac{x - 3}{2}\, dx = \frac{\int \left(x - 3\right)\, dx}{2}$$

   1. Integrate term-by-term:

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

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

      1. The integral of a constant is the constant times the variable of integration:

         $$\int \left(-3\right)\, dx = - 3 x$$

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

   So, the result is: $\frac{x^{2}}{4} - \frac{3 x}{2}$

2. Now simplify:

   $$\frac{x \left(x - 6\right)}{4}$$

3. Add the constant of integration:

   $$\frac{x \left(x - 6\right)}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(x - 6\right)}{4}+ \mathrm{constant}$$