Integral of 2x/x*sqrt((2x+1)/2x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\frac{2 x}{x} \sqrt{x \frac{2 x + 1}{2}} = \sqrt{2} \sqrt{2 x^{2} + x}$$

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

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

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

         But the integral is

         $$\int \sqrt{2 x^{2} + x}\, dx$$

      So, the result is: $\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$

   Method #2

   1. Rewrite the integrand:

      $$\frac{2 x}{x} \sqrt{x \frac{2 x + 1}{2}} = 2 \sqrt{x^{2} + \frac{x}{2}}$$

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

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

      1. Rewrite the integrand:

         $$\text{True}$$

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

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

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

            But the integral is

            $$\int \sqrt{2 x^{2} + x}\, dx$$

         So, the result is: $\frac{\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx}{2}$

      So, the result is: $\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$

2. Now simplify:

   $$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx$$

3. Add the constant of integration:

   $$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx+ \mathrm{constant}$$

The answer is:

$$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx+ \mathrm{constant}$$