Integral of 2x-y^2 dx

1. Integrate term-by-term:

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

      $$\int 2 x\, dx = 2 \int x\, dx$$

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

         $$\int x\, dx = \frac{x^{2}}{2}$$

      So, the result is: $x^{2}$

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

      $$\int \left(- y^{2}\right)\, dx = - x y^{2}$$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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