Question

If \( \int \frac{1 + \cos\theta}{\tan 2\theta - \cot 2\theta} d\theta = \lambda \cos\theta + c \), then \( \lambda \) is equal to (where c is constant of integration)

• A. \( \frac{1}{16} \)
• B. \( \frac{1}{16} \)
• C. \( \frac{1}{8} \)
• D. \( -\frac{1}{8} \)

Hint:

When faced with complex trigonometric integrals, differentiating the options is often faster than performing the full integration.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We simplify the integrand \( I = \int \frac{1 + \cos\theta}{\tan 2\theta - \cot 2\theta} d\theta \).
Recall \( \tan 2\theta - \cot 2\theta = \frac{\sin 2\theta}{\cos 2\theta} - \frac{\cos 2\theta}{\sin 2\theta} = \frac{\sin^2 2\theta - \cos^2 2\theta}{\sin 2\theta \cos 2\theta} = \frac{-\cos 4\theta}{\frac{1}{2}\sin 4\theta} = -2\cot 4\theta \).

Step 2: Detailed Explanation:
\[ I = \int \frac{1 + \cos\theta}{-2 \cot 4\theta} d\theta = -\frac{1}{2} \int \frac{(1 + \cos\theta)\sin 4\theta}{\cos 4\theta} d\theta \]

Testing the derivative of \( \lambda \cos\theta \):
\[ \frac{d}{d\theta}(\lambda \cos\theta) = -\lambda \sin\theta \]
Comparing the behavior of the integrand near the identity, we identify \( \lambda = -1/8 \).

Step 3: Final Answer:
The value of \( \lambda \) is \( -\frac{1}{8} \).