Integral of x+5/sqrt(0,5x+1)-x dx

1. Integrate term-by-term:

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

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

      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. 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. Don't know the steps in finding this integral.

         But the integral is

         $$10 \sqrt{2} \sqrt{x + 2}$$

      The result is: $\frac{x^{2}}{2} + 10 \sqrt{2} \sqrt{x + 2}$

   The result is: $10 \sqrt{2} \sqrt{x + 2}$

2. Now simplify:

   $$10 \sqrt{2 x + 4}$$

3. Add the constant of integration:

   $$10 \sqrt{2 x + 4}+ \mathrm{constant}$$

The answer is:

$$10 \sqrt{2 x + 4}+ \mathrm{constant}$$