求 $\displaystyle\int_0^1 3x^2\mathrm{e}^{3x}\,\mathrm{d}x$ 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。

先对指数函数积分，连续使用两次分部积分，使多项式的次数逐次降低。

Let $I=\int_0^1 3x^2e^{3x}\,\mathrm{d}x$. Using integration by parts gives

For the remaining integral, integrate by parts again:

Therefore

也可设原函数为指数函数乘一个二次多项式。求导并比较系数，可以直接确定这个多项式。

Let an antiderivative be $e^{3x}(ax^2+bx+c)$. Differentiating gives

Matching this to $3x^2e^{3x}$ gives

Thus $a=1$, $b=-2/3$ and $c=2/9$. Evaluating at $1$ and $0$ gives