Key Concepts and Formulas

• Differentiation of Integrals: If $\int f(x) dx = F(x) + C$, then $\frac{d}{dx} F(x) = f(x)$.
• Product Rule of Differentiation: $\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$.
• Derivatives of Trigonometric Functions: $\frac{d}{dx} \sec x = \sec x \tan x$ and $\frac{d}{dx} \tan x = \sec^2 x$.
• Integrals of Trigonometric Functions: $\int \sec x \tan x \, dx = \sec x + C$ and $\int \sec^2 x \, dx = \tan x + C$.

Step-by-Step Solution

Step 1: Write down the given equation. We are given that $\int {{e^{\sec x}}(\sec x\tan xf(x) + \sec x\tan x + \sec^2 x)dx} = {e^{\sec x}}f(x) + C$

Step 2: Differentiate both sides of the equation with respect to $x$. Differentiating both sides with respect to $x$, we have: $\frac{d}{dx} \left[\int {{e^{\sec x}}(\sec x\tan xf(x) + \sec x\tan x + \sec^2 x)dx} \right] = \frac{d}{dx} \left[{e^{\sec x}}f(x) + C\right]$ $e^{\sec x}(\sec x\tan xf(x) + \sec x\tan x + \sec^2 x) = \frac{d}{dx} \left[e^{\sec x}f(x)\right] + \frac{d}{dx} [C]$ Using the product rule for differentiation on the right-hand side, $e^{\sec x}(\sec x\tan xf(x) + \sec x\tan x + \sec^2 x) = e^{\sec x} \sec x \tan x f(x) + e^{\sec x} f'(x) + 0$

Step 3: Simplify the equation. Divide both sides by $e^{\sec x}$: $\sec x\tan xf(x) + \sec x\tan x + \sec^2 x = \sec x \tan x f(x) + f'(x)$ Subtract $\sec x \tan x f(x)$ from both sides: $\sec x\tan x + \sec^2 x = f'(x)$ Therefore, $f'(x) = \sec x\tan x + \sec^2 x$.

Step 4: Integrate $f'(x)$ to find $f(x)$. $f(x) = \int f'(x) dx = \int (\sec x\tan x + \sec^2 x) dx$ $f(x) = \int \sec x\tan x \, dx + \int \sec^2 x \, dx$ Using the standard integrals, $f(x) = \sec x + \tan x + K$ where $K$ is the constant of integration.

Step 5: Compare with the given options. The options are: (A) $x \sec x + \tan x + 1/2$ (B) $\sec x + x \tan x - 1/2$ (C) $\sec x - \tan x - 1/2$ (D) $\sec x + \tan x + 1/2$

Comparing $f(x) = \sec x + \tan x + K$ with the options, we see that option (D) has the form $\sec x + \tan x + \text{constant}$. Thus, a possible choice of $f(x)$ is $\sec x + \tan x + 1/2$.

Common Mistakes & Tips

• Careful Differentiation: Be meticulous when applying the product rule. A small error can lead to an incorrect derivative and subsequent incorrect integration.
• Remember the Constant of Integration: Always add the constant of integration when finding indefinite integrals. This constant can be crucial when matching the solution to given options.
• Check Against Options: After finding a general form of $f(x)$, compare it carefully with the available options. Look for a specific value of the constant of integration that makes your solution match one of the options.

Summary

We started with the given integral equation and differentiated both sides with respect to $x$. This allowed us to find an expression for $f'(x)$. Integrating $f'(x)$ gave us $f(x) = \sec x + \tan x + K$, where $K$ is a constant. By comparing this result with the provided options, we concluded that a possible choice for $f(x)$ is $\sec x + \tan x + 1/2$.

Final Answer The final answer is \boxed{\sec x + \tan x + 1/2}, which corresponds to option (D).