Integral of y(x+y) dy

1. Rewrite the integrand:

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

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

      $$\int y^{2}\, dy = \frac{y^{3}}{3}$$

   The result is: $\frac{x y^{2}}{2} + \frac{y^{3}}{3}$

3. Now simplify:

   $$y^{2} \left(\frac{x}{2} + \frac{y}{3}\right)$$

4. Add the constant of integration:

   $$y^{2} \left(\frac{x}{2} + \frac{y}{3}\right)+ \mathrm{constant}$$

The answer is:

$$y^{2} \left(\frac{x}{2} + \frac{y}{3}\right)+ \mathrm{constant}$$