Integral of x-6y dy

1. Integrate term-by-term:

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

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

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

      $$\int \left(- 6 y\right)\, dy = - 6 \int y\, dy$$

      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: $- 3 y^{2}$

   The result is: $x y - 3 y^{2}$

2. Now simplify:

   $$y \left(x - 3 y\right)$$

3. Add the constant of integration:

   $$y \left(x - 3 y\right)+ \mathrm{constant}$$

The answer is:

$$y \left(x - 3 y\right)+ \mathrm{constant}$$