Let f(x) be a real valued function such that `f(x)=f(121/x), AA x>0`.If `int_1^11 f(x)/xdx=5`, then the value of `int_1^121f(x)/xdx` is equal to

Answer

To solve the problem, we start with the given conditions and work through the integration step by step.

### Step 1: Understand the given function and integral
We are given that \( f(x) = f\left(\frac{121}{x}\right) \) for \( x > 0 \) and that:
\[ \int_1^{11} \frac{f(x)}{x} \, dx = 5 \]
We need to find the value of:
\[ \int_1^{121} \frac{f(x)}{x} \, dx \]

### Step 2: Break the integral into two parts
We can split the integral from 1 to 121 into two parts:
\[ \int_1^{121} \frac{f(x)}{x} \, dx = \int_1^{11} \frac{f(x)}{x} \, dx + \int_{11}^{121} \frac{f(x)}{x} \, dx \]
Let:
\[ I = \int_1^{121} \frac{f(x)}{x} \, dx \]
Then we can write:
\[ I = \int_1^{11} \frac{f(x)}{x} \, dx + \int_{11}^{121} \frac{f(x)}{x} \, dx \]
From the given information, we know:
\[ \int_1^{11} \frac{f(x)}{x} \, dx = 5 \]
So we have:
\[ I = 5 + \int_{11}^{121} \frac{f(x)}{x} \, dx \]

### Step 3: Change of variables in the second integral
For the integral from 11 to 121, we will use the property of the function \( f(x) \):
\[ \int_{11}^{121} \frac{f(x)}{x} \, dx = \int_{11}^{121} \frac{f\left(\frac{121}{x}\right)}{x} \, dx \]
Now, we will perform a change of variable. Let:
\[ x = \frac{121}{t} \quad \text{then} \quad dx = -\frac{121}{t^2} \, dt \]
When \( x = 11 \), \( t = \frac{121}{11} = 11 \) and when \( x = 121 \), \( t = 1 \). Thus, the limits change from 11 to 1.

Substituting these into the integral gives us:
\[ \int_{11}^{121} \frac{f\left(\frac{121}{x}\right)}{x} \, dx = \int_{11}^{1} \frac{f(t)}{\frac{121}{t}} \left(-\frac{121}{t^2}\right) dt \]
This simplifies to:
\[ \int_{1}^{11} \frac{f(t)}{t} \, dt \]

### Step 4: Substitute back into the integral
Thus, we find:
\[ \int_{11}^{121} \frac{f(x)}{x} \, dx = \int_{1}^{11} \frac{f(t)}{t} \, dt = 5 \]

### Step 5: Combine results
Now substituting back into our expression for \( I \):
\[ I = 5 + 5 = 10 \]

### Final Answer
Therefore, the value of \( \int_1^{121} \frac{f(x)}{x} \, dx \) is:
\[ \boxed{10} \]