Integral of cos(x/2)-x+pi dx

1. Integrate term-by-term:

   1. Integrate term-by-term:

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

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

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x\, dx = \frac{x^{2}}{2}$$

         So, the result is: $- \frac{x^{2}}{2}$

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

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

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

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

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

            1. The integral of cosine is sine:

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

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

         Now substitute $u$ back in:

         $$2 \sin{\left(\frac{x}{2} \right)}$$

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

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

      $$\int \pi\, dx = \pi x$$

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

2. Now simplify:

   $$- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}$$

3. Add the constant of integration:

   $$- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}+ \mathrm{constant}$$

The answer is: