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

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. Let $u = \sqrt{x}$.

      Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$:

      $$\int 2\, du$$

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

         $$\text{False}$$

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

            $$\int 1\, du = u$$

         So, the result is: $2 u$

      Now substitute $u$ back in:

      $$2 \sqrt{x}$$

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

2. Now simplify:

   $$\frac{2 \sqrt{x} \left(x + 3\right)}{3}$$

3. Add the constant of integration:

   $$\frac{2 \sqrt{x} \left(x + 3\right)}{3}+ \mathrm{constant}$$

The answer is: