$\int \frac{\sin (5x/2)}{\sin (x/2)}dx=$

$\begin{array}{l}\int \frac{\sin (5x/2)}{\sin (x/2)}dx=\int \frac{2\sin (5x/2)\cos (x/2)}{2\sin (x/2)\cos (x/2)}dx=\int \frac{\sin 3x+\sin 2x}{\sin x}dx\\ =\int \frac{3\sin x-4{\sin }^{3}x+2\sin x\cos x}{\sin x}dx=\int \left(3-4{\sin }^{2}x+2\cos x\right)dx\\ =\int [3-2(1-\cos 2x)+2\cos x]dx=\int (1+\cos 2x+2\cos x)dx=x+\sin 2x+2\sin x+c\end{array}$