Integral of x^2-y dy

1. Integrate term-by-term:

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

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

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

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

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

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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