Integral of sqrt2-2sin(x) dx

1. Integrate term-by-term:

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

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

      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:

      $$\int \sqrt{2}\, dx = \sqrt{2} x$$

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

2. Add the constant of integration:

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

The answer is:

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