求 $\displaystyle\int_0^\pi x^2\cos\frac13x\,\mathrm{d}x$ 的精确值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

两次分部积分依次降低多项式次数。三角函数内部的三分之一会使积分系数乘三，务必逐步保留。

Let

For the first integration by parts, take $u=x^2$ and $\mathrm{d}v=\cos(x/3)\,\mathrm{d}x$. Then $\mathrm{d}u=2x\,\mathrm{d}x$ and $v=3\sin(x/3)$, giving

For the remaining integral, take $u=x$ and $\mathrm{d}v=\sin(x/3)\,\mathrm{d}x$, so $v=-3\cos(x/3)$. Hence

Substituting this result into $I$,

At $x=0$, all terms vanish. At $x=\pi$, use $\sin(\pi/3)=\sqrt3/2$ and $\cos(\pi/3)=1/2$: