设

(a) 将 $\mathrm{f}(x)$ 分解为部分分式。

(b) 由此求 $\displaystyle\int_1^3\mathrm{f}(x)\,\mathrm{d}x$，答案以化简后的精确形式表示。

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。

二次因子的分子先设一次式。清分母后代入 $x=1/3$，再比较常数项与二次项。

Write

Then

At $x=1/3$, $28/9=A(28/9)$, so $A=1$.

Comparing constant terms gives $0=3A-C$, so $C=3$. Comparing coefficients of $x^2$ gives $1=A+3B$, so $B=0$.

Therefore

承接(a)分别积分得到对数与反正切；上下限给出两个特殊角，最后合并对数。

Using part (a), the derivatives of $\ln(3x-1)$ and $\tan^{-1}(x/\sqrt3)$ are $3/(3x-1)$ and $\sqrt3/(x^2+3)$ respectively. Their integral coefficients are therefore $1/3$ and $\sqrt3$. Also $3x-1>0$ on $1\le x\le3$. Hence