Question

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

• A. $2x+sinx+2sin2x+c$
• B. $x+2sinx+2sin2x+c$
• C. $x+2sinx+sin2x+c$
• D. $2x+sinx+sin2x+c$

Answer 1

Answer: (C)

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