Consider the function defined for in .

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Consider the function h(x)=xcos⁡xh(x)=x\cos x defined for xx in [−π,π][-\pi,\pi]. (a) Determine whether hh is an even function, an odd function, or neither. Support your conclusion algebraically. (b) Without directly evaluating any antiderivatives, predict the average value of hh on [−π,π][-\pi,\pi] and explain your reasoning. (c) Confirm your prediction by explicitly calculating 12π∫−ππxcos⁡x dx\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} x\cos x\,dx.

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