### Home > CALC > Chapter 4 > Lesson 4.1.2 > Problem4-22

4-22.

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 $\int _ { - \pi } ^ { \pi } x \cdot \operatorname { sin } x d x$.

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

2. Use a graph to justify your conclusion.

$f(a) = a\operatorname{sin}(a)$
$f(-a) = −a\operatorname{sin}(-a)$
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.

Use the eTool below to examine the graph.
Click the link at right for the full version of the eTool: Calc 4-22 HW eTool