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The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.
- Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
- Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
- Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).