If `int e^(sec x)(sec x tan x f(x)+(sec x tan x + sec^(2) x))dx = e^(sec x)f(x) + C`, then a possible choice of f(x) is

Options

A. `x sec x + tan x + (1)/(2)`
B. `sec x + tan x + (1)/(2)`
C. `sec x + x tan x -(1)/(2)`
D. `sec x - tan x -(1)/(2)`

Explanation

To solve the given problem, we start with the equation:

\[ \int e^{\sec x} \left( \sec x \tan x f(x) + \sec x \tan x + \sec^2 x \right) dx = e^{\sec x} f(x) + C \]

### Step 1: Differentiate Both Sides
We differentiate both sides of the equation with respect to \(x\).

Using the product rule on the right-hand side, we have:

\[ \frac{d}{dx} \left( e^{\sec x} f(x) \right) = e^{\sec x} f'(x) + e^{\sec x} \sec x \tan x f(x) \]

The left-hand side, using the Leibniz rule for differentiation under the integral sign, gives us:

\[ e^{\sec x} \left( \sec x \tan x f(x) + \sec x \tan x + \sec^2 x \right) \]

So, we equate both sides:

\[ e^{\sec x} \left( \sec x \tan x f(x) + \sec x \tan x + \sec^2 x \right) = e^{\sec x} f'(x) + e^{\sec x} \sec x \tan x f(x) \]

### Step 2: Simplify the Equation
We can cancel \(e^{\sec x}\) from both sides (assuming \(e^{\sec x} \neq 0\)):

\[ \sec x \tan x f(x) + \sec x \tan x + \sec^2 x = f'(x) + \sec x \tan x f(x) \]

This simplifies to:

\[ \sec x \tan x + \sec^2 x = f'(x) \]

### Step 3: Integrate Both Sides
Now, we integrate both sides with respect to \(x\):

\[ f(x) = \int \left( \sec x \tan x + \sec^2 x \right) dx \]

### Step 4: Solve the Integrals
The integral of \(\sec x \tan x\) is \(\sec x\) and the integral of \(\sec^2 x\) is \(\tan x\):

\[ f(x) = \sec x + \tan x + C \]

### Conclusion
Thus, a possible choice of \(f(x)\) is:

\[ f(x) = \sec x + \tan x + C \]