常数 $a$ 满足 $\displaystyle\int_1^a6x\ln x\,\mathrm{d}x=4$。

(a) 证明 $a=\exp\bigg(\frac16\big(\frac5{a^2}+3\big)\bigg)$，其中 $\exp(x)$ 表示 $\mathrm{e}^x$。

(b) 通过计算验证 $a$ 位于 $2$ 与 $2.1$ 之间。

(c) 使用基于 (a) 问中方程的迭代公式，求 $a$ 的值，精确到小数点后 $2$ 位。每次迭代的结果保留小数点后 $4$ 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。

分部积分后代入上下限，再把含 $\ln a$ 的项单独留下，最后指数化。

Using $u=\ln x$ and $\mathrm{d}v=6x\,\mathrm{d}x$, integration by parts gives

The given condition becomes

Multiplying by $2$ and rearranging,

Since $a>0$,

Therefore

利用(a)推得的代数对数方程构造连续函数，再检验变号。

Let

Then

The function is continuous on $[2,2.1]$, so the sign change proves that a root lies between $2$ and $2.1$.

The equation also requires $\ln a>1/2$, and for $a>1$, $f'(a)=12a\ln a>0$. Thus the root is unique, and $\boxed{2<a<2.1}$.

用(a)的重排迭代，表中保留四位小数，内部保留完整精度。

Use

with $a_0=2$. The iterations are

Hence $\boxed{a=2.02}$ to 2 decimal places.