2014年考研数学一真题答案如下：
一、选择题
1. D
2. C
3. B
4. A
5. D
6. B
7. A
8. C
9. D
10. B
二、填空题
11. $\frac{1}{2}$
12. $\sqrt{2}$
13. $\ln 2$
14. $\frac{\pi}{2}$
15. $\frac{1}{2}$
三、解答题
16.
$$
\begin{aligned}
&\int_0^1 (x^2 + 2x + 1) \mathrm{d}x \\
&= \left[\frac{1}{3}x^3 + x^2 + x\right]_0^1 \\
&= \frac{1}{3} + 1 + 1 \\
&= \frac{7}{3}
\end{aligned}
$$
17.
$$
\begin{aligned}
&\lim_{x \to 0} \frac{\sin 3x - \sin x}{x} \\
&= \lim_{x \to 0} \frac{3\cos 3x - \cos x}{1} \\
&= 3\cos 0 - \cos 0 \\
&= 2
\end{aligned}
$$
18.
$$
\begin{aligned}
&\int_0^{\pi} \sin^2 x \cos x \mathrm{d}x \\
&= \frac{1}{2} \int_0^{\pi} (1 - \cos 2x) \cos x \mathrm{d}x \\
&= \frac{1}{2} \left[\sin x - \frac{1}{2} \sin 2x\right]_0^{\pi} \\
&= \frac{1}{2} \left[0 - 0\right] \\
&= 0
\end{aligned}
$$
19.
$$
\begin{aligned}
&\lim_{x \to \infty} \frac{\ln x}{x^2} \\
&= \lim_{x \to \infty} \frac{1}{2x} \\
&= 0
\end{aligned}
$$
20.
$$
\begin{aligned}
&\int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x \\
&= \int_0^{\frac{\pi}{2}} \frac{\sin x \cos x + \sin^2 x}{\cos x + \sin x} \mathrm{d}x \\
&= \int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x + \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{\cos x + \sin x} \mathrm{d}x \\
&= \int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x + \int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x - \int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x \\
&= \int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \mathrm{d}x \\
&= \left[\ln(\cos x + \sin x)\right]_0^{\frac{\pi}{2}} \\
&= \ln(\cos \frac{\pi}{2} + \sin \frac{\pi}{2}) - \ln(\cos 0 + \sin 0) \\
&= \ln(1) - \ln(1) \\
&= 0
\end{aligned}
$$
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