It seems like you're asking to evaluate the following integral:

$\int 4 \sqrt{x} \, dx$

Let's solve this step by step.

Step 1: Express $\sqrt{x}$ as $x^{1/2}$

We rewrite the integral as:

$\int 4 x^{1/2} \, dx$

Step 2: Apply the power rule for integration

The power rule of integration states that:

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

where $n \neq -1$.

In our case, $n = \frac{1}{2}$, so we apply the power rule:

$\int 4 x^{1/2} \, dx = 4 \cdot \frac{x^{(1/2) + 1}}{(1/2) + 1} + C$

Simplifying the exponent and the denominator:

$= 4 \cdot \frac{x^{3/2}}{3/2} + C$

Step 3: Simplify the expression

Now, simplify the constant term:

$4 \cdot \frac{2}{3} = \frac{8}{3}$

So, the integral becomes:

$\frac{8}{3} x^{3/2} + C$

Final Answer:

$\int 4 \sqrt{x} \, dx = \frac{8}{3} x^{3/2} + C$

Would you like further details or have any questions about this? Here are some related questions you may find interesting:

1. How do you handle integrals with square roots or fractional powers?
2. What happens if the exponent is negative in a similar integral?
3. How would you solve a definite integral with limits for this problem?
4. Can you integrate other functions like $\int x^n \, dx$ when $n$ is a fraction or negative?
5. What are the steps for solving an integral involving a constant multiple, like $\int 5 \sqrt{x} \, dx$?

Tip: Always remember to apply the power rule for integration, and when dealing with fractional exponents, treat them just like any other exponent for ease.