To solve the problem, we need to evaluate the integral \( \int_0^2 f(x) \, dx \) given that \( f(x) \) is a differentiable function on the interval \( (0, 2) \).
### Step-by-Step Solution:
1. **Define a New Function**:
Let \( H(t) = \int_0^t f(x) \, dx \) for \( t \in [0, 2] \). This function \( H(t) \) represents the area under the curve \( f(x) \) from 0 to \( t \).
**Hint**: This step helps us to relate the integral to a function that we can analyze using calculus.
2. **Evaluate at the Endpoints**:
We calculate \( H(2) \) and \( H(0) \):
- \( H(2) = \int_0^2 f(x) \, dx \)
- \( H(0) = \int_0^0 f(x) \, dx = 0 \)
**Hint**: The integral from 0 to 0 is always zero, which simplifies our calculations.
3. **Apply the Mean Value Theorem (MVT)**:
According to the Mean Value Theorem for integrals, there exists a \( c \in (0, 2) \) such that:
\[
H'(c) = \frac{H(2) - H(0)}{2 - 0}
\]
This simplifies to:
\[
H'(c) = \frac{H(2)}{2}
\]
**Hint**: The Mean Value Theorem connects the average rate of change of a function to its derivative at some point in the interval.
4. **Find the Derivative of \( H(t) \)**:
By the Fundamental Theorem of Calculus, we know that:
\[
H'(t) = f(t)
\]
Therefore, substituting \( c \) into the equation gives us:
\[
f(c) = \frac{H(2)}{2}
\]
**Hint**: Understanding that the derivative of the integral gives back the original function is key to applying the Fundamental Theorem of Calculus.
5. **Relate Back to the Integral**:
From the equation \( f(c) = \frac{H(2)}{2} \), we can express \( H(2) \) as:
\[
H(2) = 2f(c)
\]
Thus, we have:
\[
\int_0^2 f(x) \, dx = 2f(c)
\]
where \( c \in (0, 2) \).
**Hint**: This step shows that the value of the integral is directly related to the value of the function at some point in the interval.
### Final Result:
The value of \( \int_0^2 f(x) \, dx \) is given by \( 2f(c) \) for some \( c \in (0, 2) \).
### Conclusion:
The final answer is \( \int_0^2 f(x) \, dx = 2f(c) \) where \( c \) is some point in the interval \( (0, 2) \).
