Let t f(x)= int x log_(1/2)(1 - 2 ^ (- t)) dt and I= f(2) +1(2-log) then which of the following is

The problem involves an integral that defines a function $ f(x) $ and an expression $ I = f(2) + 1(2 - \log) $. However, there seem to be some details missing, such as the options for the question and the complete expression for the logarithm term in $ I $.

Let's clarify the situation and solve the problem as much as possible:

### Given:
\[
f(x) = \int x \log_{\frac{1}{2}}\left(1 - 2^{-t}\right) \, dt
\]
And
\[
I = f(2) + 1 \cdot (2 - \log)
\]

### Step 1: Understanding $ f(x) $
The function $ f(x) $ is defined as an integral. However, the limits of the integral are missing, and the expression for $ I $ is also incomplete. Typically, the integral would have definite limits, and the logarithm would have a specific base and argument.

### Step 2: Analyze the Expression for $ I $
Given the equation:
\[
I = f(2) + 1 \cdot (2 - \log)
\]
It seems that $ \log $ is meant to have an argument, likely something like $ \log_2(3) $ or similar. Without the exact argument, we can't fully simplify $ I $.

### Step 3: Solve for $ f(2) $
To proceed, we need to determine $ f(2) $. If we assume the integral has no limits and evaluate $ f(x) $ at $ x = 2 $, we would proceed by calculating the integral or recognizing if there's a standard result that applies.

However, since the integral is complicated and the exact form of $ I $ isn't clear, I'll need more information to solve this correctly.

### Conclusion
If you provide the exact options or more details about the logarithm term, I can analyze which one is correct.

Would you like to clarify these details or provide the multiple-choice options? 

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**Relative Questions:**
1. What are the limits of the integral in the definition of $ f(x) $?
2. What is the exact argument and base of the logarithm in the expression $ I $?
3. How do we typically evaluate indefinite integrals involving logarithmic functions?
4. Could the logarithmic base $ \frac{1}{2} $ impact the result significantly?
5. What methods can be used to approximate integrals when the exact evaluation is difficult?

**Tip:** When dealing with logarithms, especially with unusual bases like $ \frac{1}{2} $, it's essential to remember that the properties of logarithms may lead to different interpretations and results compared to more common bases like 2 or 10.

Explore the integral function $f(x)$ defined by $f(x) = \int x \log_{\frac{1}{2}}\left(1 - 2^{-t}\right) dt$ and its relevance in evaluating the expression $I = f(2) + 1(2 - \log)$. Dive into the complexities of logarithmic functions and definite integrals in advanced mathematics.

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