If ` int sec 2 x dx = f[ g(x)] + C`, then

Options

A. `f(x) = log |x|, g(x) = tan ((x)/(4) -x)`
B. `f(x) = log |x|, g(x) = cot((x)/(4) -x)`
C. `f(x) = (1)/(2) log |x|, g(x) = tan ((x)/(4) -x)`
D. `f(x) = (1)/(2) log |x| , g(x) = cot((pi)/(4) -x)`

Answer

To solve the integral \( \int \sec^2 x \, dx \) and express it in the form \( f(g(x)) + C \), we will follow these steps:

### Step 1: Integrate \( \sec^2 x \)
The integral of \( \sec^2 x \) is a standard result:
\[ \int \sec^2 x \, dx = \tan x + C \]

### Step 2: Express the result in the form \( f(g(x)) + C \)
We need to express \( \tan x \) in the form \( f(g(x)) \). To do this, we can let:
- \( g(x) = x \)
- \( f(x) = \tan x \)

Thus, we can write:
\[ \tan x = f(g(x)) \]

### Step 3: Identify \( f(x) \) and \( g(x) \)
From our expressions, we have:
- \( f(x) = \tan x \)
- \( g(x) = x \)

### Final Result
Thus, we have:
\[ \int \sec^2 x \, dx = f(g(x)) + C = \tan(g(x)) + C = \tan(x) + C \]

### Summary of Results
- \( f(x) = \tan x \)
- \( g(x) = x \)