Question

If \(\int_0^{\pi/2} \log(\cos x) \, dx = \frac{\pi}{2} \log\left(\frac{1}{2}\right)\), then find \(\int_0^{\pi/2} \log(\sec x) \, dx\).

Answer 1

Step 1: Use the identity for sec(x):
We are tasked with solving the integral: \[ \int_0^{\frac{\pi}{2}} \log(\sec(x)) \, dx \] We begin by using the trigonometric identity \( \sec(x) = \frac{1}{\cos(x)} \). This allows us to rewrite the integral as: \[ \int_0^{\frac{\pi}{2}} \log\left(\frac{1}{\cos(x)}\right) \, dx \]

Step 2: Simplify the logarithm:
We now apply the property of logarithms that \( \log\left(\frac{1}{a}\right) = -\log(a) \), so the integral becomes: \[ \int_0^{\frac{\pi}{2}} -\log(\cos(x)) \, dx \]

Step 3: Use the known value for the integral of \( \log(\cos(x)) \):
From known integral results, we have: \[ \int_0^{\frac{\pi}{2}} \log(\cos(x)) \, dx = -\frac{\pi}{2} \log(2) \] Therefore: \[ -\int_0^{\frac{\pi}{2}} \log(\cos(x)) \, dx = \frac{\pi}{2} \log(2) \]

Step 4: Substitute the result into the original integral:
Substituting this result back into the original integral: \[ \int_0^{\frac{\pi}{2}} \log(\sec(x)) \, dx = \frac{\pi}{2} \log(2) \]

Final Answer:
Therefore, the value of the integral is: \[ \int_0^{\frac{\pi}{2}} \log(\sec(x)) \, dx = \frac{\pi}{2} \log(2) \]

Answer 2

Given:\(\int_0^{\pi/2} \log(\cos x) \, dx = \frac{\pi}{2} \log\left(\frac{1}{2}\right)\)

We know that \(\log(\sec x) = -\log(\cos x)\) because \(\sec x = \frac{1}{\cos x}\). Therefore,

\(\int_0^{\pi/2} \log(\sec x) \, dx = \int_0^{\pi/2} -\log(\cos x) \, dx = -\int_0^{\pi/2} \log(\cos x) \, dx\)

Using the given value for \(\int_0^{\pi/2} \log(\cos x) \, dx\), we substitute it into the expression:

\(-\int_0^{\pi/2} \log(\cos x) \, dx = -\left( \frac{\pi}{2} \log\left(\frac{1}{2}\right) \right)\)

\(-\left( \frac{\pi}{2} \log\left(\frac{1}{2}\right) \right) = \frac{\pi}{2} \log(2)\)

So, the answer is: \(\frac{\pi}{2} \log(2)\)