If `int_(0)^((pi)/(2))logcosxdx=(pi)/(2)log((1)/(2))`, then `int_(0)^((pi)/(2))logsecdx=`

Options

A. `(pi)/(2)log((1)/(2))`
B. `1-(pi)/(2)log((1)/(2))`
C. `1+(pi)/(2)log((1)/(2))`
D. `(pi)/(2)log2`

Explanation

To solve the problem, we need to evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \log \sec x \, dx \) given that \( \int_{0}^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left( \frac{1}{2} \right) \).

### Step-by-Step Solution:

1. **Rewrite the Integral**:
We know that \( \sec x = \frac{1}{\cos x} \). Therefore, we can rewrite the integral:
\[ \int_{0}^{\frac{\pi}{2}} \log \sec x \, dx = \int_{0}^{\frac{\pi}{2}} \log \left( \frac{1}{\cos x} \right) \, dx \]

2. **Use Logarithmic Properties**:
Using the property of logarithms \( \log \left( \frac{1}{a} \right) = -\log a \), we can simplify 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. **Substitute the Given Value**:
From the problem statement, we know that:
\[ \int_{0}^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left( \frac{1}{2} \right) \]
Substituting this value into our expression gives:
\[ -\int_{0}^{\frac{\pi}{2}} \log \cos x \, dx = -\frac{\pi}{2} \log \left( \frac{1}{2} \right) \]

4. **Simplify the Expression**:
The logarithm can be simplified:
\[ -\frac{\pi}{2} \log \left( \frac{1}{2} \right) = -\frac{\pi}{2} \left( -\log 2 \right) = \frac{\pi}{2} \log 2 \]

5. **Final Result**:
Therefore, we conclude that:
\[ \int_{0}^{\frac{\pi}{2}} \log \sec x \, dx = \frac{\pi}{2} \log 2 \]