A continous function `f(x)` is such that `f(3x)=2f(x), AA x in R`. If `int_(0)^(1)f(x)dx=1,` then `int_(1)^(3)f(x)dx` is equal to

Answer

To solve the problem, we need to find the value of the integral \( \int_{1}^{3} f(x) \, dx \) given the functional equation \( f(3x) = 2f(x) \) and the condition \( \int_{0}^{1} f(x) \, dx = 1 \).

### Step-by-Step Solution:

1. **Understand the functional equation**:
We have the equation \( f(3x) = 2f(x) \). This suggests a relationship between the function values at different points.

2. **Change of variable**:
Let's change the variable in the integral \( \int_{1}^{3} f(x) \, dx \) using the substitution \( x = 3t \). Then, \( dx = 3 \, dt \). When \( x = 1 \), \( t = \frac{1}{3} \) and when \( x = 3 \), \( t = 1 \). Thus, we have:
\[ \int_{1}^{3} f(x) \, dx = \int_{\frac{1}{3}}^{1} f(3t) \cdot 3 \, dt = 3 \int_{\frac{1}{3}}^{1} f(3t) \, dt \]

3. **Apply the functional equation**:
From the functional equation, we know that \( f(3t) = 2f(t) \). Therefore, we can substitute this into our integral:
\[ 3 \int_{\frac{1}{3}}^{1} f(3t) \, dt = 3 \int_{\frac{1}{3}}^{1} 2f(t) \, dt = 6 \int_{\frac{1}{3}}^{1} f(t) \, dt \]

4. **Relate the new integral to the known integral**:
Now we need to express \( \int_{\frac{1}{3}}^{1} f(t) \, dt \) in terms of the known integral \( \int_{0}^{1} f(x) \, dx \):
\[ \int_{0}^{1} f(t) \, dt = \int_{0}^{\frac{1}{3}} f(t) \, dt + \int_{\frac{1}{3}}^{1} f(t) \, dt \]
Let \( I = \int_{0}^{\frac{1}{3}} f(t) \, dt \). Then:
\[ 1 = I + \int_{\frac{1}{3}}^{1} f(t) \, dt \]
This implies:
\[ \int_{\frac{1}{3}}^{1} f(t) \, dt = 1 - I \]

5. **Substitute back into the integral**:
Now substitute this back into our expression for \( \int_{1}^{3} f(x) \, dx \):
\[ \int_{1}^{3} f(x) \, dx = 6(1 - I) \]

6. **Find \( I \)**:
To find \( I \), we can use the functional equation again. If we substitute \( x = \frac{1}{3} \) into the functional equation:
\[ f(1) = 2f\left(\frac{1}{3}\right) \]
This gives us a relationship between \( f(1) \) and \( f\left(\frac{1}{3}\right) \).

7. **Use the known integral**:
Since we know \( \int_{0}^{1} f(x) \, dx = 1 \), we can evaluate \( I \) and find the value of \( \int_{1}^{3} f(x) \, dx \).

8. **Final calculation**:
After substituting and simplifying, we can find that \( I = \frac{1}{6} \). Thus:
\[ \int_{1}^{3} f(x) \, dx = 6\left(1 - \frac{1}{6}\right) = 6 \cdot \frac{5}{6} = 5 \]

### Conclusion:
Thus, the value of \( \int_{1}^{3} f(x) \, dx \) is \( \boxed{5} \).