If `int_(0)^(pi//2) log cos x dx =(pi)/(2)log ((1)/(2)),` then `int_(0)^(pi//2) log sec x dx =`

Options

A. `(pi)/(2)log ""((1)/(2))`
B. `1-(pi)/(2)log ""((1)/(2))`
C. `1+(pi)/(2) log ""((1)/(2))`
D. `(pi)/(2) log 2`

Answer

To solve the problem, we need to find the value of the integral \( \int_0^{\frac{\pi}{2}} \log \sec x \, dx \) given that \( \int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \frac{1}{2} \).

### Step-by-step Solution:

1. **Understanding the relationship between sec and cos:**
We know that \( \sec x = \frac{1}{\cos x} \). Therefore, we can express \( \log \sec x \) in terms of \( \log \cos x \):
\[ \log \sec x = \log \left( \frac{1}{\cos x} \right) = -\log \cos x \]

2. **Setting up the integral:**
Now we can rewrite the integral:
\[ \int_0^{\frac{\pi}{2}} \log \sec x \, dx = \int_0^{\frac{\pi}{2}} -\log \cos x \, dx = -\int_0^{\frac{\pi}{2}} \log \cos x \, dx \]

3. **Substituting the known value:**
We substitute the known value of the integral \( \int_0^{\frac{\pi}{2}} \log \cos x \, dx \):
\[ -\int_0^{\frac{\pi}{2}} \log \cos x \, dx = -\left( \frac{\pi}{2} \log \frac{1}{2} \right) \]

4. **Simplifying the expression:**
We can simplify \( -\frac{\pi}{2} \log \frac{1}{2} \):
\[ -\frac{\pi}{2} \log \frac{1}{2} = -\frac{\pi}{2} (-\log 2) = \frac{\pi}{2} \log 2 \]

5. **Final result:**
Thus, we find that:
\[ \int_0^{\frac{\pi}{2}} \log \sec x \, dx = \frac{\pi}{2} \log 2 \]

### Conclusion:
The value of the integral \( \int_0^{\frac{\pi}{2}} \log \sec x \, dx \) is \( \frac{\pi}{2} \log 2 \).

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