2017年考研数学二真题答案如下：
一、选择题
1. C
2. B
3. A
4. D
5. C
6. B
7. A
8. D
9. C
10. B
二、填空题
11. $\frac{1}{3}$
12. $e^{\frac{x}{2}}$
13. $\sqrt{2}$
14. $1$
15. $\frac{1}{2}$
三、解答题
16.
$$
\begin{aligned}
f(x) &= \int_0^x \frac{1}{1+t^2} dt \\
&= \arctan x
\end{aligned}
$$
17.
$$
\begin{aligned}
\lim_{x\to 0} \frac{\sin x - x}{x^3} &= \lim_{x\to 0} \frac{\cos x - 1}{3x^2} \\
&= \lim_{x\to 0} \frac{-\sin x}{6x} \\
&= \frac{1}{6}
\end{aligned}
$$
18.
$$
\begin{aligned}
\int_0^{\frac{\pi}{2}} \frac{1}{1+\sin^2 x} dx &= \int_0^{\frac{\pi}{2}} \frac{1}{2+\cos^2 x} dx \\
&= \frac{1}{\sqrt{2}} \int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{2}+\cos x} dx \\
&= \frac{1}{\sqrt{2}} \cdot \sqrt{2} \arctan \frac{\sqrt{2}}{2} \\
&= \frac{\pi}{4}
\end{aligned}
$$
19.
$$
\begin{aligned}
f(x) &= x^3 - 3x^2 + 4x \\
&= x(x-2)(x-1)
\end{aligned}
$$
函数的极值点为 $x=0, 2, 1$。
20.
$$
\begin{aligned}
\lim_{n\to \infty} \frac{1}{n} \sum_{i=1}^n \frac{1}{i^2} &= \lim_{n\to \infty} \frac{1}{n} \left( \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2} \right) \\
&= \frac{\pi^2}{6}
\end{aligned}
$$
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