We are tasked with solving the integral:

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

Let's break this into two parts:

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

Solving the first part:

$\int \frac{1}{x} \, dx$

This is a basic logarithmic integral:

$\int \frac{1}{x} \, dx = \ln |x|$

Solving the second part:

$\int \sqrt{x} \, dx = \int x^{1/2} \, dx$

We apply the power rule for integration:

$\int x^n \, dx = \frac{x^{n+1}}{n+1}$

Here, $n = \frac{1}{2}$, so:

$\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$

Combining the two results:

$\int \left( \frac{1}{x} + \sqrt{x} \right) \, dx = \ln |x| + \frac{2}{3} x^{3/2} + C$

Now, matching this with the options given, the correct choice is:

$\boxed{\text{ln}(x) + \frac{2}{3} x^{3/2} + c}$

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the power rule for integration work, and when can it be applied?
2. Why is $\int \frac{1}{x} \, dx = \ln |x|$, and why does the absolute value matter?
3. Can you explain what happens when integrating negative exponents?
4. How can integrals with square roots be handled in more complex situations?
5. What are some applications of this type of integral in physics or engineering?

Tip: Always check if you can split integrals when adding functions—it simplifies the process!