The integral provided in the image is:

$\int \frac{1}{\sqrt{x+2x}} \, dx$

Let me simplify and solve it step-by-step:

Step 1: Simplify the expression

In the denominator: $x + 2x = 3x$ So the integral becomes: $\int \frac{1}{\sqrt{3x}} \, dx$

Step 2: Rewrite the integral

Factor out the constants: $\int \frac{1}{\sqrt{3} \sqrt{x}} \, dx = \frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x}} \, dx$

Step 3: Solve the integral

The integral of $\frac{1}{\sqrt{x}}$ is $2\sqrt{x}$: $\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x}} \, dx = \frac{1}{\sqrt{3}} \cdot 2\sqrt{x} + C$

Step 4: Simplify the result

$\frac{2\sqrt{x}}{\sqrt{3}} + C$

This is the final answer.

Do you have any questions, or would you like further clarification? Here are five follow-up questions you might find useful:

1. What is the derivation for the integral of $\frac{1}{\sqrt{x}}$?
2. How can we verify the solution by differentiating it?
3. What happens if the integral contains a different constant inside the square root?
4. Can this method be generalized for integrals of the form $\int \frac{1}{\sqrt{ax}} dx$?
5. How does the constant of integration $C$ affect the solution?

Tip: Always check for simplifications in the integral before solving to make the process easier.