Integral of (-((x*y+1)/x)+y/x) dy

1. Integrate term-by-term:

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

      $$\int \frac{y}{x}\, dy = \frac{\int y\, dy}{x}$$

      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: $\frac{y^{2}}{2 x}$

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

      $$\int \left(- \frac{x y + 1}{x}\right)\, dy = - \frac{\int \left(x y + 1\right)\, dy}{x}$$

      1. Integrate term-by-term:

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

            $$\int x y\, dy = x \int y\, dy$$

            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: $\frac{x y^{2}}{2}$

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

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

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

      So, the result is: $- \frac{\frac{x y^{2}}{2} + y}{x}$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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