Question

If \[ \int \frac{\sin x}{\sin(x - \alpha)} dx = Ax + B \log \sin(x - \alpha) + C, \] \(\text{then the value of}\) \( (A, B) \) is:

• A. \( (-\cos \alpha, \sin \alpha) \)
• B. \( (\cos \alpha, \sin \alpha) \)
• C. \( (-\sin \alpha, \cos \alpha) \)
• D. \( (\sin \alpha, \cos \alpha) \)

Hint:

Use standard integration techniques and compare terms to find constants \( A \) and \( B \) in the indefinite integral.

Answer 1

The Correct Option is C

By differentiating the given equation and comparing it with the integrand, we find that \( A = -\sin \alpha \) and \( B = \cos \alpha \).
Final Answer: \[ \boxed{(-\sin \alpha, \cos \alpha)} \]