Integral of sin(0.5*x) dx

1. Let $u = \frac{x}{2}$.

   Then let $du = \frac{dx}{2}$ and substitute $2 du$:

   $$\int 4 \sin{\left(u \right)}\, du$$

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

      $$\int 2 \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$$

      1. The integral of sine is negative cosine:

         $$\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$$

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

   Now substitute $u$ back in:

   $$- 2 \cos{\left(\frac{x}{2} \right)}$$

2. Add the constant of integration:

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

The answer is:

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