Integral of (2*sqrt(x)-sqrt(2x)+5) 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:

         $$\int 2 \sqrt{x}\, dx = 2 \int \sqrt{x}\, dx$$

         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: $\frac{4 x^{\frac{3}{2}}}{3}$

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

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

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

            But the integral is

            $$\frac{2 \sqrt{2} x^{\frac{3}{2}}}{3}$$

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

      The result is: $- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3}$

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

      $$\int 5\, dx = 5 x$$

   The result is: $- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x$

2. Add the constant of integration:

   $$- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x+ \mathrm{constant}$$

The answer is: