Integral of (2x+2)dy-(2y) dx

1. Integrate term-by-term:

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

      $$\int \left(- 2 y\right)\, dy = - 2 \int y\, dy$$

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

         $$\int y\, dy = \frac{y^{2}}{2}$$

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

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

      $$\int \left(2 x + 2\right)\, dy = y \left(2 x + 2\right)$$

   The result is: $- y^{2} + y \left(2 x + 2\right)$

2. Now simplify:

   $$y \left(2 x - y + 2\right)$$

3. Add the constant of integration:

   $$y \left(2 x - y + 2\right)+ \mathrm{constant}$$

The answer is:

$$y \left(2 x - y + 2\right)+ \mathrm{constant}$$