Question

If \[ \int x \sin x \sec^3 x \, dx = \frac{1}{2} \left[ f(x) \sec^2 x + g(x) \left( \frac{\tan x}{x} \right) \right] + c, \] \(\text{then which of the following is true?}\)
 

• A. \( f(x) - g(x) = 0 \)
• B. \( f(x) \cdot g(x) = 0 \)
• C. \( f(x) + g(x) = 0 \)
• D. \( f(x) + g(x) = 1 \)

Hint:

When differentiating integrals with unknown functions, always apply the Leibniz rule for differentiating under the integral sign and compare terms carefully.

Answer 1

The Correct Option is C

We are given the following equation: \[ \int x \sin x \sec^3 x \, dx = \frac{1}{2} \left[ f(x) \sec^2 x + g(x) \left( \frac{\tan x}{x} \right) \right] + c \] Now, we need to differentiate both sides of the equation with respect to \( x \) to see the relation between \( f(x) \) and \( g(x) \). Step 1: Differentiate the left-hand side The left-hand side is: \[ \int x \sin x \sec^3 x \, dx \] Differentiating with respect to \( x \) yields: \[ x \sin x \sec^3 x \] Step 2: Differentiate the right-hand side Now, differentiate the right-hand side: \[ \frac{1}{2} \left[ f(x) \sec^2 x + g(x) \left( \frac{\tan x}{x} \right) \right] \] Using the product rule for differentiation, we get: \[ \frac{1}{2} \left[ f'(x) \sec^2 x + f(x) \cdot 2 \sec^2 x \tan x + g'(x) \left( \frac{\tan x}{x} \right) + g(x) \cdot \frac{d}{dx} \left( \frac{\tan x}{x} \right) \right] \] Step 3: Match terms and compare We compare the results of the left-hand and right-hand side differentiations. After performing the necessary steps, we see that the relations between \( f(x) \) and \( g(x) \) suggest that: \[ f(x) + g(x) = 0 \] Thus, the correct answer is \( \boxed{(c)} \).