(i) 求

并把答案写成最简形式。

(ii) 使用部分分式与代数积分，证明

其中 $k$ 是一个需要求出的最简有理常数。

连续两次分部积分，将多项式次数逐步降低，最后提出共同因子 $e^{4x}$。

Let $I=\int x^2e^{4x}\,\mathrm{d}x$. Use integration by parts with

Then $\mathrm{d}u/\mathrm{d}x=2x$ and $v=e^{4x}/4$, so

where $J=\int xe^{4x}\,\mathrm{d}x$.

For $J$, take $u=x$ and $\mathrm{d}v/\mathrm{d}x=e^{4x}$. Then $\mathrm{d}u/\mathrm{d}x=1$, and

Substitute into $I$:

用表格记录连续求导与积分，以交错符号合并各行乘积；这是两次分部积分的表格路线。

Differentiate the polynomial repeatedly and integrate the exponential repeatedly:

The last derivative is zero, so there is no remaining integral. Using the alternating signs,

先用部分分式分解再积分。代入上下限时正确处理绝对值，用对数律合并并化简有理常数。

Write the partial fractions as

Multiplying by the denominator gives the identity

At $x=2$, $15=5B$, so $B=3$. At $x=-1/2$, $10=5A/2$, so $A=4$.

Thus the integrand is $4/(2x+1)+3/(2-x)$. An antiderivative is

For $4\le x\le7$, $2x+1>0$ and $2-x<0$, so the logarithms are defined using the displayed absolute values.

At the endpoints,

The required integral is $F(7)-F(4)$. Combining the corresponding logarithms,

Hence $k=8/45$, a fully simplified rational constant.