Integral of 3sqrtx-(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:

      $$\int 3 \sqrt{x}\, dx = 3 \int \sqrt{x}\, dx$$

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

         $$\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$$

      So, the result is: $2 x^{\frac{3}{2}}$

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

      $$\int \left(- (x + 2)\right)\, dx = - \int \left(x + 2\right)\, dx$$

      1. Integrate term-by-term:

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

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

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

            $$\int 2\, dx = 2 x$$

         The result is: $\frac{x^{2}}{2} + 2 x$

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

   The result is: $2 x^{\frac{3}{2}} - \frac{x^{2}}{2} - 2 x$

2. Add the constant of integration:

   $$2 x^{\frac{3}{2}} - \frac{x^{2}}{2} - 2 x+ \mathrm{constant}$$

The answer is:

$$2 x^{\frac{3}{2}} - \frac{x^{2}}{2} - 2 x+ \mathrm{constant}$$