Let `f(0)=0a n dint_0^2f^(prime)(2t)e^(f(2t))dt=5.t h e nv a l u eoff(4)i s` log 2 (b) log 7 (c) log 11 (d) log 13

Options

A. `log 2`
B. `log 7`
C. `log 11`
D. `log 13`

Explanation

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step-by-Step Solution:

1. **Given Information**:
We have the following information:
- \( f(0) = 0 \)
- \( \int_0^2 f'(2t) e^{f(2t)} dt = 5 \)

2. **Substitution**:
We will use the substitution method. Let:
\[ y = e^{f(2t)} \]
Then, differentiating both sides gives:
\[ dy = f'(2t) e^{f(2t)} \cdot 2 dt \quad \Rightarrow \quad dt = \frac{dy}{2 f'(2t) e^{f(2t)}} \]

3. **Changing Limits**:
When \( t = 0 \):
\[ y = e^{f(0)} = e^0 = 1 \]
When \( t = 2 \):
\[ y = e^{f(4)} \]

4. **Rewriting the Integral**:
The integral can be rewritten as:
\[ \int_1^{e^{f(4)}} \frac{1}{2} dy = 5 \]
This simplifies to:
\[ \frac{1}{2} \left( e^{f(4)} - 1 \right) = 5 \]

5. **Solving for \( e^{f(4)} \)**:
Multiplying both sides by 2 gives:
\[ e^{f(4)} - 1 = 10 \]
Therefore:
\[ e^{f(4)} = 11 \]

6. **Finding \( f(4) \)**:
Taking the natural logarithm on both sides:
\[ f(4) = \log(11) \]

### Conclusion:
The value of \( f(4) \) is \( \log(11) \).

### Final Answer:
The correct option is (c) \( \log 11 \).