Question

Find : $\int \sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)dx$

Answer 1

$\int \sin \frac{x}{2}\cdot \cos \frac{x}{2}dx$

Multiply by $\frac{2}{2}$

$\frac{2}{2}\int \sin \frac{x}{2}\cos \frac{x}{2}dx$

$\int 2\sin \frac{x}{2}\cos \frac{x}{2}dx$

$\frac{1}{2}\int \sin xdx$

$[\because \sin x=2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)]$

$=-\frac{\cos x}{2}$.