Let `f(x)` be a function satisying `f(x) = f((100)/(x)) AA x ge 0`. If `int_(1)^(10) (f(x))/(x) dx = 5` then find the value of `int_(1)^(100)(f(x))/(x) dx`.

Answer

To solve the problem, we need to evaluate the integral \( \int_{1}^{100} \frac{f(x)}{x} \, dx \) given that \( f(x) = f\left(\frac{100}{x}\right) \) for \( x \geq 0 \) and \( \int_{1}^{10} \frac{f(x)}{x} \, dx = 5 \).

### Step-by-Step Solution:

1. **Split the Integral**:
We can split the integral from 1 to 100 into two parts:
\[ \int_{1}^{100} \frac{f(x)}{x} \, dx = \int_{1}^{10} \frac{f(x)}{x} \, dx + \int_{10}^{100} \frac{f(x)}{x} \, dx \]

2. **Change of Variable in the Second Integral**:
For the second integral \( \int_{10}^{100} \frac{f(x)}{x} \, dx \), we will use the substitution \( x = \frac{100}{t} \). Then, \( dx = -\frac{100}{t^2} \, dt \).
- When \( x = 10 \), \( t = \frac{100}{10} = 10 \).
- When \( x = 100 \), \( t = \frac{100}{100} = 1 \).

Therefore, the limits of integration change from \( x = 10 \) to \( x = 100 \) into \( t = 10 \) to \( t = 1 \):
\[ \int_{10}^{100} \frac{f(x)}{x} \, dx = \int_{10}^{1} \frac{f\left(\frac{100}{t}\right)}{\frac{100}{t}} \left(-\frac{100}{t^2}\right) dt \]
This simplifies to:
\[ = \int_{1}^{10} \frac{f\left(\frac{100}{t}\right)}{t} dt \]

3. **Using the Property of the Function**:
From the property \( f(x) = f\left(\frac{100}{x}\right) \), we can replace \( f\left(\frac{100}{t}\right) \) with \( f(t) \):
\[ \int_{10}^{100} \frac{f(x)}{x} \, dx = \int_{1}^{10} \frac{f(t)}{t} dt \]

4. **Combine the Integrals**:
Now we can combine our results:
\[ \int_{1}^{100} \frac{f(x)}{x} \, dx = \int_{1}^{10} \frac{f(x)}{x} \, dx + \int_{1}^{10} \frac{f(t)}{t} dt \]
Since both integrals are equal:
\[ \int_{1}^{100} \frac{f(x)}{x} \, dx = \int_{1}^{10} \frac{f(x)}{x} \, dx + \int_{1}^{10} \frac{f(x)}{x} \, dx = 2 \int_{1}^{10} \frac{f(x)}{x} \, dx \]

5. **Substituting the Given Value**:
We know from the problem statement that \( \int_{1}^{10} \frac{f(x)}{x} \, dx = 5 \):
\[ \int_{1}^{100} \frac{f(x)}{x} \, dx = 2 \times 5 = 10 \]

### Final Answer:
Thus, the value of \( \int_{1}^{100} \frac{f(x)}{x} \, dx \) is \( \boxed{10} \).