Integral of 1/12*(x+2y) dy

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

   $$\int \frac{x + 2 y}{12}\, dy = \frac{\int \left(x + 2 y\right)\, dy}{12}$$

   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:

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

         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: $y^{2}$

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

   So, the result is: $\frac{x y}{12} + \frac{y^{2}}{12}$

2. Now simplify:

   $$\frac{y \left(x + y\right)}{12}$$

3. Add the constant of integration:

   $$\frac{y \left(x + y\right)}{12}+ \mathrm{constant}$$

The answer is:

$$\frac{y \left(x + y\right)}{12}+ \mathrm{constant}$$