Question

\( \int \frac{(\sin x + \cos x)(2 - \sin 2x)}{\sin^2 2x} \, dx \)

• A. \( \frac{\sin x + \cos x}{\sin 2x} + c \)
• B. \( \frac{\sin x - \cos x}{\sin 2x} + c \)
• C. \( \frac{\cos x}{\sin x} + c \)
• D. \( \frac{\sin x}{\cos x} + c \)
• E. \( \frac{1}{\sin 2x} + c \)

Hint:

Look for expressions whose derivative partially appears in numerator.

Answer 1

The Correct Option is A

Concept: Use substitution and trig identities.

Step 1: Recall: \[ \sin 2x = 2\sin x \cos x \]

Step 2: Substitute: \[ \sin^2 2x = 4\sin^2 x \cos^2 x \]

Step 3: Simplify integrand.

Step 4: Let: \[ u = \sin x + \cos x \]

Step 5: Then: \[ du = (\cos x - \sin x) dx \]

Step 6: Rewrite integrand in terms of \(u\).

Step 7: Integrate simplified form.

Step 8: Final result: \[ \frac{\sin x + \cos x}{\sin 2x} + c \]