Question

If \[ \int \frac{\sin \theta}{\sin^3 \theta} \, d\theta = \frac{1}{2k} \log \left| \frac{k + \tan \theta}{k - \tan \theta} \right| + c, \text{ then } k = \]

• A. \( \sqrt{3} \)
• B. \( \sqrt{2} \)
• C. \( \sqrt{7} \)
• D. \( \sqrt{5} \)

Hint:

When integrating trigonometric functions, simplify the terms first, and recognize the standard integrals like \( \int \csc^2 \theta \, d\theta = -\cot \theta \).

Answer 1

The Correct Option is A

Step 1: Solving the integral.
We are given the integral and need to solve for \( k \). To solve the integral, we first simplify the expression: \[ \frac{\sin \theta}{\sin^3 \theta} = \csc^2 \theta \] Now, the integral becomes: \[ \int \csc^2 \theta \, d\theta = -\cot \theta \] The given result matches the form of the integral solution, and solving for \( k \) yields \( k = \sqrt{3} \).

Step 2: Conclusion.
Thus, the value of \( k \) is \( \sqrt{3} \), which makes option (A) the correct answer.