https://file.aiquickdraw.com/mathbot/file/1726252741554fjkofzyo.jpg

The integral shown in the image is:

\[
I = \int \frac{\sqrt{1 - 4\sqrt{x}}}{\sqrt{x}} \, dx
\]

with the limits of integration from 1 to 3.

Let me solve this step by step.

### Step 1: Substitution
We notice that there is a nested square root, so we can make a substitution to simplify it. Let's try:

\[
u = \sqrt{x} \implies du = \frac{1}{2\sqrt{x}} dx \quad \text{or} \quad dx = 2u \, du
\]

This transforms the integral into:

- \( \sqrt{x} = u \)
- \( \sqrt{1 - 4\sqrt{x}} = \sqrt{1 - 4u} \)
- The limits change from \( x = 1 \to u = 1 \) and \( x = 3 \to u = \sqrt{3} \).

So, the integral becomes:

\[
I = 2 \int_1^{\sqrt{3}} \frac{\sqrt{1 - 4u}}{u} u \, du
\]

The \( u \)'s cancel, simplifying it to:

\[
I = 2 \int_1^{\sqrt{3}} \sqrt{1 - 4u} \, du
\]

### Step 2: New substitution
Now, let's attempt another substitution to handle the square root:

Let \( v = 1 - 4u \), then:

- \( dv = -4 \, du \)
- When \( u = 1 \), \( v = -3 \)
- When \( u = \sqrt{3} \), \( v = 1 - 4\sqrt{3} \).

I will now compute the transformed integral.The integral of \(\sqrt{1 - 4u}\) with respect to \(u\) is:

\[
\int \sqrt{1 - 4u} \, du = -\frac{(1 - 4u)^{3/2}}{6}
\]

### Step 3: Applying limits
We now evaluate this expression from \( u = 1 \) to \( u = \sqrt{3} \).

Let me calculate the result of applying these limits.

I = ∫ (sqrt(1 - 4sqrt(x)) / sqrt(x)) dx from x=1 to x=3

Solve the integral ∫ (sqrt(1 - 4sqrt(x)) / sqrt(x)) dx with limits 1 to 3. This solution involves substitution methods and the evaluation of definite integrals, suitable for students learning advanced calculus techniques.

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