If `int_(0)^(pi//2) log(cosx) 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 \( \int_{0}^{\frac{\pi}{2}} \log(\cos x) \, dx = \frac{\pi}{2} \log\left(\frac{1}{2}\right) \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
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. **Using Logarithmic Properties**:
Using the property of logarithms \( \log\left(\frac{1}{a}\right) = -\log(a) \), we can simplify:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = \int_{0}^{\frac{\pi}{2}} -\log(\cos x) \, dx \]
This can be expressed as:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = -\int_{0}^{\frac{\pi}{2}} \log(\cos x) \, dx \]

3. **Substituting the Given Value**:
We know from the problem statement that:
\[ \int_{0}^{\frac{\pi}{2}} \log(\cos x) \, dx = \frac{\pi}{2} \log\left(\frac{1}{2}\right) \]
Therefore, substituting this value into our equation gives:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = -\left(\frac{\pi}{2} \log\left(\frac{1}{2}\right)\right) \]

4. **Simplifying the Expression**:
We can simplify \( \log\left(\frac{1}{2}\right) \):
\[ \log\left(\frac{1}{2}\right) = -\log(2) \]
Thus, we have:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = -\left(\frac{\pi}{2} \cdot -\log(2)\right) = \frac{\pi}{2} \log(2) \]

5. **Final Result**:
Therefore, the value of the integral \( \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx \) is:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = \frac{\pi}{2} \log(2) \]

### Conclusion:
The final answer is:
\[ \int_{0}^{\frac{\pi}{2}} \log(\sec x) \, dx = \frac{\pi}{2} \log(2) \]