Integral of sqrt(x)-x+2 dx

1. Integrate term-by-term:

   1. Integrate term-by-term:

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

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

      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: $- \frac{x^{2}}{2}$

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

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

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

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

2. Add the constant of integration:

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

The answer is:

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