Question

Find the domain of \( f(x) \) given: \[ \int \frac{\sin x \cos x}{\sqrt{\cos^4 x - \sin^4 x}} dx = -\frac{f(x)}{2} + C. \] then domain of f{x) is

• A. \( [2n\pi, (2n+1)\pi] \), \( n = 0, 1, 2, \dots \)
• B. \( [(4n - 1) \frac{\pi}{2}, (4n+1) \frac{\pi}{2}] \), \( n = 0, 1, 2, \dots \)
• C. \( [(4n - 1) \frac{\pi}{4}, (4n+1) \frac{\pi}{4}] \), \( n = 0, 1, 2, \dots \)
• D. \( [(2n \frac{\pi}{4}, (2n+1) \frac{\pi}{4}] \), \( n = 0, 1, 2, \dots \)

Hint:

When determining the domain, find values where the denominator or square root term becomes zero or undefined.

Answer 1

The Correct Option is C

Step 1: Analyze the denominator The given integral has the denominator: \[ \sqrt{\cos^4 x - \sin^4 x}. \] Using trigonometric identities, \[ \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = \cos 2x. \] 

Step 2: Domain restrictions The function is undefined where the denominator is zero: \[ \cos 2x = 0 \Rightarrow 2x = \frac{\pi}{2} + n\pi. \] Solving for \( x \), \[ x = \frac{\pi}{4} + n\frac{\pi}{2}. \] Thus, the domain of \( f(x) \) is: \[ [(4n - 1) \frac{\pi}{4}, (4n+1) \frac{\pi}{4}]. \]