Step 1: Understanding the Concept:
This integral can be solved using the method of substitution (u-substitution). The key is to identify a part of the integrand whose derivative is also present in the integrand.
Step 2: Key Formula or Approach:
We will use the substitution method. Let $u = \log(\sec x)$. Then we find $du$ in terms of $dx$ and see if it matches the rest of the integrand. We will need the derivative formula $\frac{d}{dx}(\log u) = \frac{1}{u}\frac{du}{dx}$ and $\frac{d}{dx}(\sec x) = \sec x \tan x$.
Step 3: Detailed Explanation:
The integral is $\int (\log \sec x) \tan x \,dx$.
Let's choose the substitution:
\[ u = \log(\sec x) \]
Now, let's find the derivative of $u$ with respect to $x$:
\[ \frac{du}{dx} = \frac{d}{dx}(\log(\sec x)) \]
Using the chain rule:
\[ \frac{du}{dx} = \frac{1}{\sec x} \cdot \frac{d}{dx}(\sec x) \]
\[ \frac{du}{dx} = \frac{1}{\sec x} \cdot (\sec x \tan x) \]
\[ \frac{du}{dx} = \tan x \]
This means we can write $du = \tan x \,dx$.
Now, substitute $u$ and $du$ back into the original integral:
\[ \int (\log \sec x) \tan x \,dx = \int u \,du \]
This is a simple power rule integration:
\[ \int u \,du = \frac{u^2}{2} + C \]
Finally, substitute back $u = \log(\sec x)$:
\[ = \frac{(\log \sec x)^2}{2} + C \]
Step 4: Final Answer:
The value of the integral is $\frac{1}{2}(\log \sec x)^2 + c$. Therefore, option (C) is correct.