(a) 求 $8x^3+4x^2+2x+7$ 除以 $4x^2+1$ 的商和余数。

(b) 由此求

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。

依次除最高次项，再减去对应倍数的除式，剩下常数余式。

Subtracting $2x(4x^2+1)$,

Subtracting one more copy of the divisor gives

Therefore

The quotient is $\boxed{2x+1}$ and the remainder is $\boxed{6}$.

承接(a)，将被积函数写成一次多项式加反正切型分式，逐项积分。

Using part (a),

Since $\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(2x)=2/(1+4x^2)$, the remaining integral is $3\tan^{-1}(2x)$. Thus