Question

Evaluate
${\int }_{\pi /2}^{\pi }\frac{\sin \left(3x/2\right)}{\sin \left(x/2\right)}dx$

Answer 1

Now,

$\begin{array}{c}={\int }_{\pi /2}^{\pi }\frac{\sin \left(\frac{3x}{2}\right)}{\sin \left(\frac{x}{2}\right)}dx\\ ={\int }_{\pi /2}^{\pi }\left\{\frac{\sin .cos\frac{x}{2}}{\sin \left(\frac{x}{2}\right)}+\frac{\cos x.\sin x\frac{x}{2}}{\sin \left(\frac{x}{2}\right)}\right\}dx\\ ={\int }_{\pi /2}^{\pi }\left\{\frac{2\sin \frac{x}{2}.{\cos }^2\frac{x}{2}}{\sin \left(\frac{x}{2}\right)}+\cos x\right\}dx\\ =2{\int }_{\pi /2}^{\pi }\left(\frac{1+\cos x}{2}\right)dx+{\int }_{\pi /2}^{\pi }\cos xdx\\ ={\left[x+\sin x+\sin x\right]}_{\pi /2}^{\pi }\\ =\left(\pi +0+0\right)-\left(\frac{\pi }{2}+1+1\right)\\ =\frac{\pi }{2}-2\end{array}$