Question

Evaluate the integral with square root in denominator. \[ \int \frac{\sin 2x \cos 2x}{\sqrt{9-\cos^4 2x}}\,dx \]

• A. \(\frac{1}{4} \sin^{-1}\left(\frac{\cos^2 2x}{2}\right)\)
• B. \(\frac{-1}{4} \sin^{-1}\left(\frac{\cos^2 2x}{2}\right)\)
• C. \(\frac{1}{2} \sin^{-1}\left(\frac{\cos^2 2x}{2}\right)\)
• D. \(\frac{-1}{2} \sin^{-1}\left(\frac{\cos^2 2x}{2}\right)\)
• E. None of these

Hint:

For any integration question, always check that the full integral is present:
• integrand,
• variable,
• limits if definite. e} If the actual integral is missing, the answer cannot be found uniquely.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Substitution leads to a standard integral form.
Step 2: Key Formula or Approach:
Put \(t = \cos^2 2x \implies dt = -4 \sin 2x \cos 2x dx\).
Step 3: Detailed Explanation:
Integral becomes \(-1/4 \int \frac{dt}{\sqrt{9-t^2}}\). (Assuming const is \(4\) based on options).
If const is \(4\), then \(-1/4 \sin^{-1}(t/2)\).
Step 4: Final Answer:
Matches option (B).