Integral of -2x+8 dx

1. Integrate term-by-term:

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

      $$\int 8\, dx = 8 x$$

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

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

      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: $- x^{2} + 8 x$

2. Now simplify:

   $$x \left(8 - x\right)$$

3. Add the constant of integration:

   $$x \left(8 - x\right)+ \mathrm{constant}$$

The answer is:

$$x \left(8 - x\right)+ \mathrm{constant}$$