<p>$\int{x\,\sin }\left( \frac{x}{2} \right)\,\mathrm{d}x$</p>

$\displaystyle\displaystyle \int{x\,\sin }\left( \frac{x}{2} \right)\,\mathrm{d}x$

Techniques of  Integration - General

Techniques of  Integration

International Baccalaureate Higher Level Mathematics

$\displaystyle\int x \sin\left(\frac{x}{2}\right) dx$

Using LIATE: $u = x$, $\frac{\mathrm{d}v}{\mathrm{d}x} = \sin\left(\frac{x}{2}\right)$

$\frac{\mathrm{d}u}{\mathrm{d}x} = 1$, $v = -\frac{\cos(\frac{x}{2})}{\frac{1}{2}} = -2\cos\left(\frac{x}{2}\right)$

$= x\left(-2\cos\left(\frac{x}{2}\right)\right) - \displaystyle\int -2\cos\left(\frac{x}{2}\right) dx$

$= -2x\cos\left(\frac{x}{2}\right) + 2 \cdot \frac{\sin(\frac{x}{2})}{\frac{1}{2}} + C$

$= -2x\cos\left(\frac{x}{2}\right) + 4\sin\left(\frac{x}{2}\right) + C$