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To solve the given problem, we need to evaluate the integral 

\[ \int_0^1 \frac{f(x)}{\sqrt{x}} \, dx \]

where 

\[ f(x) = \int_1^x \frac{\ln(t+1)}{t} \, dt. \]

### Step-by-Step Solution

1. **Substitute \( f(x) \) into the given integral:**

   \[ \int_0^1 \frac{f(x)}{\sqrt{x}} \, dx = \int_0^1 \frac{1}{\sqrt{x}} \left( \int_1^x \frac{\ln(t+1)}{t} \, dt \right) dx. \]

2. **Interchange the order of integration:**

   To interchange the order of integration, we need to determine the region of integration. The limits of the outer integral are from 0 to 1, and for each \( x \) in this range, the inner integral is from 1 to \( x \). 

   By interchanging the order of integration, the region changes. The new region for \( t \) will be from 1 to 1 and for \( x \) will be from \( t \) to 1. Thus,

   \[ \int_0^1 \frac{1}{\sqrt{x}} \left( \int_1^x \frac{\ln(t+1)}{t} \, dt \right) dx = \int_1^1 \frac{\ln(t+1)}{t} \left( \int_t^1 \frac{1}{\sqrt{x}} \, dx \right) dt. \]

3. **Evaluate the inner integral:**

   \[ \int_t^1 \frac{1}{\sqrt{x}} \, dx = \left[ 2\sqrt{x} \right]_t^1 = 2\sqrt{1} - 2\sqrt{t} = 2 - 2\sqrt{t}. \]

4. **Substitute the result back into the outer integral:**

   \[ \int_1^1 \frac{\ln(t+1)}{t} (2 - 2\sqrt{t}) \, dt. \]

   Split the integral into two parts:

   \[ 2 \int_1^1 \frac{\ln(t+1)}{t} \, dt - 2 \int_1^1 \frac{\ln(t+1)}{t} \sqrt{t} \, dt. \]

5. **Simplify each part separately:**

   First part:
   \[ 2 \int_1^1 \frac{\ln(t+1)}{t} \, dt. \]

   Second part:
   \[ -2 \int_1^1 \frac{\ln(t+1)}{t} \sqrt{t} \, dt. \]

   Since \( t \) is in the range from 1 to 1, the above integrals simplify by considering the limits:

   \[ \int_1^1 \text{anything} \, dt = 0. \]

6. **Combine the results:**

   \[ 2 \cdot 0 - 2 \cdot 0 = 0. \]

Therefore, the value of the given integral is 

\[ \int_0^1 \frac{f(x)}{\sqrt{x}} \, dx = 0. \]

Would you like more details on any specific part or have any questions?

### 5 Questions to Consider:
1. How do we determine the region of integration when changing the order of integration?
2. What techniques can be used to evaluate more complex nested integrals?
3. How can the Fundamental Theorem of Calculus be applied to solve similar problems?
4. What are some common functions where the order of integration can be easily changed?
5. What role do the limits of integration play in simplifying the problem?

### Tip:
When changing the order of integration, carefully sketch the region of integration to visualize the new limits for the inner and outer integrals.

Learn how to solve the integral ∫₀¹(f(x)/√x)dx, where f(x) = ∫₁ˣ(ln(t+1)/t)dt, step-by-step. This problem involves the Fundamental Theorem of Calculus and is suitable for college-level students.

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