Question

If \[ \int \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R}, \text{ then} \] \[ \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R} \]

• A. \( f(x) = 2\sin x, A = \cos \alpha \)
• B. \( f(x) = 2\sin x, A = 2\cos \alpha \)
• C. \( f(x) = \sin x, A = \cos \alpha \)
• D. \( f(x) = \sin x, A = 2\cos \alpha \)

Hint:

To solve integrals involving trigonometric identities, consider simplifying using standard identities and methods like integration by parts or substitution.

Answer 1

The Correct Option is C

The given integral can be simplified using standard integration techniques. Upon solving the integral, we find that \( f(x) = \sin x \) and \( A = \cos \alpha \). Thus, the correct answer is option (3).