Integral of x²(y+2x)dy dx

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

   $$\int x^{2} \left(2 x + y\right)\, dy = x^{2} \int \left(2 x + y\right)\, dy$$

   1. Integrate term-by-term:

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

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

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

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

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

   So, the result is: $x^{2} \left(2 x y + \frac{y^{2}}{2}\right)$

2. Now simplify:

   $$\frac{x^{2} y \left(4 x + y\right)}{2}$$

3. Add the constant of integration:

   $$\frac{x^{2} y \left(4 x + y\right)}{2}+ \mathrm{constant}$$

The answer is: