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

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

   Hint (a):

   Is $f\left(x\right)=x\sin\left(x\right)$ even, odd or neither?

2. Use a graph to justify your conclusion.

   Hint (b):

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