Question

\[ \int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} \, dx \] is equal to :

• A. $2(\sin x + x \cos \alpha) + C$
• B. $2(\sin x - x \cos \alpha) + C$
• C. $2(\sin x + 2x \cos \alpha) + C$
• D. $2(\sin x + \sin \alpha) + C$

Hint:

For integrals involving trigonometric functions, use trigonometric identities and integration by parts to simplify the expressions.

Answer 1

The Correct Option is B

The given integral can be simplified using trigonometric identities and integration by parts. The first step is to apply the trigonometric identity: \[ \cos 2x - \cos 2\alpha = -2 \sin\left(x + \alpha\right) \sin(x - \alpha) \] Next, apply integration by parts to the resulting expression, leading to the final simplified result: \[ 2(\sin x - x \cos \alpha) + C \] Thus, the correct answer is $2(\sin x - x \cos \alpha) + C$.