$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx$ is equal to

(where c is a constant of integration.)

Let, $I=\int \frac{\sin \left(\frac{5x}{2}\right)}{\sin \frac{x}{2}}dx=\int \frac{2\sin \left(\frac{5x}{2}\right)\cos \left(\frac{x}{2}\right)}{2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)}dx$

$=\int \frac{\sin \left(\frac{5x}{2}+\frac{x}{2}\right)+\sin \left(\frac{5x}{2}-\frac{x}{2}\right)}{\sin x}dx=\int \frac{\sin 3x+\sin 2x}{\sin x}dx$

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