常数 $a$ 满足 $\displaystyle\int_1^a x^2\ln x\,\mathrm{d}x=4$。

(a) 证明

(b) 通过计算验证 $a$ 位于 $2.4$ 与 $2.8$ 之间。

(c) 利用基于 (a) 小问中方程的迭代公式求 $a$，精确到小数点后 2 位。写出每次迭代的结果，精确到小数点后 4 位。

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。

用分部积分求出定积分，再整理成 $a^3(3\ln a-1)=35$，最后取实立方根。

Integrating by parts,

Using the given integral,

Multiplying by $9$,

The original logarithm requires $a>0$, so this relation also gives $3\ln a-1>0$. Hence

将(a)右侧减去左侧，计算区间两端的符号。所选区间中立方根表达式连续。

Let

Then

The function is continuous on $[2.4,2.8]$, so a root lies between $2.4$ and $2.8$.

The original integral is strictly increasing for $a>1$, since its derivative is $a^2\ln a>0$ there. For $0<a<1$, the integral is at most $1/9$, so it cannot equal $4$. Hence this is the required unique value of $a$.

按(a)给出的公式迭代，保留内部精度，继续到相邻数值明确支持两位小数答案。

Use

The iterates are:

Therefore $\boxed{a=2.64}$ to 2 decimal places.