∫ (sin(5x)/(2))/(sin(x)/(2))dx , is equal to

We have, $I = ∫ \dfrac{sin\dfrac{5x}{2}}{sin\dfrac{x}{2}}dx$

$= ∫ \dfrac{\sin \left(2x + \dfrac{x}{2}\right)}{sin\dfrac{x}{2}}dx$

Using $\sin \left(A + B\right) = sinAcosB + cosAsinB$

$= ∫ \dfrac{\sin 2xcos\dfrac{x}{2} + \cos 2xsin\dfrac{x}{2}}{sin\dfrac{x}{2}}dx$

$= ∫ \left(\dfrac{\left(2\sin x \cos x\right)cos\dfrac{x}{2}}{sin\dfrac{x}{2}} + \cos 2x\right)dx$

$= ∫ \left(\dfrac{2\left(2\sin \left(\dfrac{x}{2}\right)\cos \left(\dfrac{x}{2}\right)\right)\cos x \cos\left(\dfrac{x}{2}\right)}{sin\dfrac{x}{2}} + \cos 2x\right)dx$

$= ∫ \left(4\cos^{2}\left(\dfrac{x}{2}\right)\cos x + \cos 2x\right)dx$

Using $\cos 2x = 2\cos^{2}x - 1 ⇒ 2\cos^{2}x = 1 + \cos 2x$

$= ∫ \left(2\left(1 + \cos x\right)\cos x + \cos 2x\right)dx$

$= ∫ \left(2\cos x + 2\cos^{2}x + \cos 2x\right)dx$

$= ∫ \left(2\cos x + \left(1 + \cos 2x\right) + \cos 2x\right)dx$

$= ∫ \left(2\cos x + 2\cos 2x + 1\right)dx$

$= 2\sin x + \sin 2x + x + c$, where $c$ is the constant of integration.