Integral of 2x-4x dx

1. Integrate term-by-term:

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

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

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

         $$\int 4 x\, dx = 4 \int x\, dx$$

         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: $2 x^{2}$

      So, the result is: $- 2 x^{2}$

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

      $$\int 2 x\, dx = 2 \int x\, dx$$

      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: $- x^{2}$

2. Add the constant of integration:

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

The answer is:

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