If `inte^(secx) (secxtanxf(x)+secxtanx+tan^(2)x)dx=e^(secx)f(x)+c`. Then `f(x)` is: (A) `secx+xtamx+1/2` (B) `xsecx+tanx+1/2` (C) `xsecx+x^2tanx+1/2` (D) `secx+tanx+1/2`

Options

A. `secx+xtanx+(1)/(2)`
B. `xsecx+tanx+(1)/(2)`
C. `xsecx+x^(2)tanx+(1)/(2)`
D. `secx+tanx+(1)/(2)`

Answer

To solve the given problem, we need to find the function \( f(x) \) such that:

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

### Step 1: Differentiate both sides
Instead of integrating, we can differentiate both sides of the equation. This will help us eliminate the integral and find \( f(x) \).

Differentiating the left-hand side using the product rule gives us:

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

### Step 2: Apply the product rule
Using the product rule, we differentiate \( e^{\sec x} \) and the expression in the parentheses:

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

Thus, applying the product rule:

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

### Step 3: Differentiate the inside expression
Now we differentiate the expression inside the parentheses:

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

Using the product rule again on \( \sec x \tan x f(x) \):

\[ \sec x \tan x f'(x) + f(x) \frac{d}{dx}(\sec x \tan x) + \frac{d}{dx}(\sec x \tan x) + \frac{d}{dx}(\sec^2 x) \]

### Step 4: Simplify the expression
The derivatives of \( \sec x \tan x \) and \( \sec^2 x \) are known:

\[ \frac{d}{dx}(\sec x \tan x) = \sec x \tan^2 x + \sec^3 x \]
\[ \frac{d}{dx}(\sec^2 x) = 2 \sec^2 x \tan x \]

### Step 5: Set the derivatives equal
Now we set the differentiated left-hand side equal to the differentiated right-hand side:

\[ e^{\sec x} \sec x \tan x \left( \sec x \tan x f(x) + \sec x \tan x + \sec^2 x \right) + e^{\sec x} \left( \sec x \tan x f'(x) + f(x)(\sec x \tan^2 x + \sec^3 x) + (\sec x \tan^2 x + \sec^3 x) + 2 \sec^2 x \tan x \right) = e^{\sec x} f'(x) \]

### Step 6: Cancel \( e^{\sec x} \)
We can cancel \( e^{\sec x} \) from both sides:

\[ \sec x \tan x \left( \sec x \tan x f(x) + \sec x \tan x + \sec^2 x \right) + \sec x \tan x f'(x) + f(x)(\sec x \tan^2 x + \sec^3 x) + (\sec x \tan^2 x + \sec^3 x) + 2 \sec^2 x \tan x = f'(x) \]

### Step 7: Solve for \( f(x) \)
From the equation, we can isolate \( f'(x) \) and integrate to find \( f(x) \):

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

Integrating gives:

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

### Conclusion
Thus, the function \( f(x) \) is:

\[ f(x) = \sec x + \tan x + \frac{1}{2} \]

### Final Answer
The correct option is (D) \( \sec x + \tan x + \frac{1}{2} \).

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