Integral of 2y-x+1 dy

1. Integrate term-by-term:

   1. Integrate term-by-term:

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

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

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

         $$\int 2 y\, dy = 2 \int y\, dy$$

         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: $y^{2}$

      The result is: $- x y + y^{2}$

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

      $$\int 1\, dy = y$$

   The result is: $- x y + y^{2} + y$

2. Now simplify:

   $$y \left(- x + y + 1\right)$$

3. Add the constant of integration:

   $$y \left(- x + y + 1\right)+ \mathrm{constant}$$

The answer is:

$$y \left(- x + y + 1\right)+ \mathrm{constant}$$