A continuous real function `f` satisfies `f(2x)=3(f(x)) AA x in R`. If `int_0^1 f(x)dx=1,` then find the value of `int_1^2f(x)dx`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given functional equation
We have the functional equation:
\[ f(2x) = 3f(x) \]
for all \( x \in \mathbb{R} \).

### Step 2: Use the integral condition
We know that:
\[ \int_0^1 f(x) \, dx = 1 \]

### Step 3: Substitute in the integral
We can substitute \( f(2x) \) into the integral:
\[ \int_0^1 f(2x) \, dx = \int_0^1 3f(x) \, dx \]
This implies:
\[ \int_0^1 f(2x) \, dx = 3 \int_0^1 f(x) \, dx = 3 \cdot 1 = 3 \]

### Step 4: Change of variable in the integral
Now, we change the variable in the integral \( \int_0^1 f(2x) \, dx \). Let \( t = 2x \), then \( dt = 2dx \) or \( dx = \frac{dt}{2} \). The limits change accordingly:
- When \( x = 0 \), \( t = 0 \)
- When \( x = 1 \), \( t = 2 \)

Thus, we have:
\[ \int_0^1 f(2x) \, dx = \int_0^2 f(t) \cdot \frac{1}{2} \, dt = \frac{1}{2} \int_0^2 f(t) \, dt \]

### Step 5: Set up the equation
From the previous steps, we have:
\[ \frac{1}{2} \int_0^2 f(t) \, dt = 3 \]
Multiplying both sides by 2 gives:
\[ \int_0^2 f(t) \, dt = 6 \]

### Step 6: Split the integral
Now we can split the integral from 0 to 2:
\[ \int_0^2 f(t) \, dt = \int_0^1 f(t) \, dt + \int_1^2 f(t) \, dt \]
Substituting the known value:
\[ 6 = 1 + \int_1^2 f(t) \, dt \]

### Step 7: Solve for the integral from 1 to 2
Rearranging gives:
\[ \int_1^2 f(t) \, dt = 6 - 1 = 5 \]

### Conclusion
Thus, the value of \( \int_1^2 f(x) \, dx \) is:
\[ \boxed{5} \]