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Let's analyze the problem and identify the correct basic integration formula and the value for \( u \).

### The given integral:
\[
\int \frac{16}{\sqrt{x(5 - 2\sqrt{x})}} \, dx
\]

### We are asked to select a basic integration formula and identify \( u \).

#### Step 1: Simplify the expression
The term inside the square root is \( \sqrt{x(5 - 2\sqrt{x})} \). Let's set up the substitution to reduce it to a standard form.

We notice the presence of a square root \( \sqrt{x} \). Let's attempt a substitution \( u = \sqrt{x} \), which means \( x = u^2 \) and \( dx = 2u \, du \).

#### Step 2: Apply the substitution
Substitute \( u = \sqrt{x} \) into the integral:

- \( x = u^2 \)
- \( dx = 2u \, du \)

Now the expression \( \sqrt{x(5 - 2\sqrt{x})} \) becomes:

\[
\sqrt{u^2 (5 - 2u)} = u \sqrt{5 - 2u}
\]

The integral becomes:

\[
\int \frac{16}{u \sqrt{5 - 2u}} \cdot 2u \, du
\]

This simplifies to:

\[
32 \int \frac{du}{\sqrt{5 - 2u}}
\]

#### Step 3: Identify the correct formula
Now the integral is of the form:

\[
\int \frac{du}{\sqrt{a^2 - u^2}}
\]

where \( a^2 = 5 \) and \( u \) corresponds to \( u \). This is a standard integral, and the basic integration formula we need is:

\[
\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C
\]

Thus, the correct formula from the options is:

\[
\boxed{\int \frac{du}{\sqrt{a^2 - u^2}}}
\]

### Step 4: Identify \( u \)
From the substitution earlier, we chose \( u = \sqrt{x} \). Therefore, the correct identification is:

\[
\boxed{u = \sqrt{x}}
\]

---

Would you like further details or clarification? Here are some related questions for further understanding:

1. How do you apply trigonometric substitution in integrals?
2. What are common substitution strategies for integrals involving square roots?
3. How can the arcsine function arise from integrals of this form?
4. What are some other cases where substitution simplifies integrals?
5. How does one identify the right substitution technique for different integrals?

**Tip**: When simplifying integrals, always look for patterns involving basic identities or common substitution strategies to reduce complexity.

Select the basic integration formula you can use to find the integral: ∫(16/√(x(5 - 2√x))) dx, and identify u.

Learn how to solve the integral ∫(16/√(x(5 - 2√x))) dx by identifying the correct substitution u = √x and applying the formula ∫ du / √(a^2 - u^2). A detailed solution with step-by-step guidance.

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