Integral of dx/1-cosx dx

1. Integrate term-by-term:

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

      $$\int 1.0\, dx = 1.0 x$$

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

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

      1. The integral of cosine is sine:

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

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

   The result is: $1.0 x - \sin{\left(x \right)}$

2. Now simplify:

   $$1.0 x - 1.0 \sin{\left(x \right)}$$

3. Add the constant of integration:

   $$1.0 x - 1.0 \sin{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$1.0 x - 1.0 \sin{\left(x \right)}+ \mathrm{constant}$$