6．Figure 1 shows a sketch of part of the curve with equation \(y = x \sin ( \ln x ) , x \geqslant 1\) \begin{figure}[h] \end{figure} For \(x > 1\) ，the curve first crosses the \(x\)－axis at the point \(A\) ．
（a）Find the \(x\) coordinate of \(A\) ．
（b）Differentiate \(x \sin ( \ln x )\) and \(x \cos ( \ln x )\) with respect to \(x\) and hence find $$\int \sin ( \ln x ) \mathrm { d } x \text { and } \int \cos ( \ln x ) \mathrm { d } x$$ （c）（i）Find \(\int x \sin ( \ln x ) \mathrm { d } x\) ．
（ii）Hence show that the area of the shaded region \(\boldsymbol { R }\) ，bounded by the curve and the \(x\)－axis between the points \(( 1,0 )\) and \(A\) ，is $$\frac { 1 } { 5 } \left( \mathrm { e } ^ { 2 \pi } + 1 \right)$$