求 $\displaystyle\int_0^1x\tan^{-1}x\,\mathrm{d}x$ 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。

分部积分时对反正切求导、对 $x$ 积分。余下分式先拆成一与简单分式，再积分。

Let $u=\tan^{-1}x$ and $\mathrm{d}v=x\,\mathrm{d}x$. Integration by parts gives

The lower-limit contributions are zero because $x=0$ and $\tan^{-1}0=0$. Hence