The value of $\int _ { 0 } ^ { \pi } x \cdot \operatorname { sin } x d x$ is $π$.  

1. Without a calculator, find: $2 \int _ { 0 } ^ { \pi } x \cdot \operatorname { sin } x d x$ and $\int _ { - \pi } ^ { \pi } x \cdot \operatorname { sin } x d x$.

   Hint (a):

   Is $f(x) = x\operatorname{sin}(x)$ even, odd or neither?

2. Use a graph to justify your conclusion.

   Hint (b):

   $f(a) = a\operatorname{sin}(a)$
   $f(-a) = −a\operatorname{sin}(-a)$
   but, since $y = \operatorname{sin}x$ is odd,
   $= −a(−\operatorname{sin}(a)) = a\operatorname{sin}(a)$
   So $f(a) = f(−a)$
   therefore, $y = x\operatorname{sin}x$ is even.