Question

$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} \, dx =$

• A. $2x + \sin x + 2\sin 2x + C$
• B. $x + 2\sin x + 2\sin 2x + C$
• C. $x + 2\sin x + \sin 2x + C$
• D. $2x + \sin x + \sin 2x + C$

Answer 1

The Correct Option is C

The given integral is: \[ I = \int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} \, dx. \] Using the trigonometric identity: \[ \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} = 2\cos \frac{2x}{2} + \cos x. \] Substitute this into the integral: \[ I = \int \left(2\cos x + \cos x\right) dx = \int 2\cos x \, dx + \int \cos x \, dx. \] The integrals simplify as follows: \[ \int 2\cos x \, dx = 2\sin x, \quad \int \cos x \, dx = \sin 2x. \] Combine the results: \[ I = x + 2\sin x + \sin 2x + C. \]