Integral of (2+2cosx) dx

1. Integrate term-by-term:

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

      $$\int 2 \cos{\left(x \right)}\, dx = 2 \int \cos{\left(x \right)}\, dx$$

      1. The integral of cosine is sine:

         $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

      So, the result is: $2 \sin{\left(x \right)}$

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

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

   The result is: $2 x + 2 \sin{\left(x \right)}$

2. Add the constant of integration:

   $$2 x + 2 \sin{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$2 x + 2 \sin{\left(x \right)}+ \mathrm{constant}$$