Integral of 2sinx-1 dx

1. Integrate term-by-term:

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

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

      1. The integral of sine is negative cosine:

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

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

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

      $$\int \left(\left(-1\right) 1\right)\, dx = - x$$

   The result is: $- x - 2 \cos{\left(x \right)}$

2. Add the constant of integration:

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

The answer is:

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