Integral of 4x-1/2*sqrt(x-2) dx

1. Integrate term-by-term:

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

      $$\int 4 x\, dx = 4 \int x\, dx$$

      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: $2 x^{2}$

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

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

      1. Let $u = x - 2$.

         Then let $du = dx$ and substitute $du$:

         $$\int \sqrt{u}\, du$$

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

            $$\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$$

         Now substitute $u$ back in:

         $$\frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}$$

      So, the result is: $- \frac{\left(x - 2\right)^{\frac{3}{2}}}{3}$

   The result is: $2 x^{2} - \frac{\left(x - 2\right)^{\frac{3}{2}}}{3}$

2. Now simplify:

   $$2 x^{2} - \frac{\left(x - 2\right)^{\frac{3}{2}}}{3}$$

3. Add the constant of integration:

   $$2 x^{2} - \frac{\left(x - 2\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$$

The answer is:

$$2 x^{2} - \frac{\left(x - 2\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$$