Question

$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx$ is equal to:
(where $c$ is a constant of integration.)

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

Answer 1

The correct option is A $x+2\sin x+\sin 2x+c$

$I=\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx$
$=\int \frac{2\sin \frac{5x}{2}\cos \frac{x}{2}}{\sin x}dx$
$=\int \frac{\sin 3x+\sin 2x}{\sin x}dx$
$=\int \frac{(3\sin x-4{\sin }^{3}x)+2\sin x\cos x}{\sin x}dx$
$=\int (3-4{\sin }^{2}x+2\cos x)dx$
$=\int (3-2(1-\cos 2x)+2\cos x)dx$
$=\int (3-2+2\cos 2x+2\cos x)dx$
$=\int (1+2\cos 2x+2\cos x)dx$
$=x+\sin 2x+2\sin x+c$