考研数学常用公式速查

一、导数

二、积分

1. 基本积分表

• $\int k \, dx = kx + C$

• $\int x^a \, dx = \frac{x^{a+1}}{a+1} + C \quad (a \neq -1)$

• $\int \frac{1}{x} \, dx = \ln|x| + C$

• $\int a^x \, dx = \frac{a^x}{\ln a} + C$

• $\int e^x \, dx = e^x + C$

2. 三角函数积分

• $\int \cos x \, dx = \sin x + C$

• $\int \sin x \, dx = -\cos x + C$

• $\int \sec^2 x \, dx = \tan x + C$

• $\int \csc^2 x \, dx = -\cot x + C$

• $\int \tan x \, dx = -\ln|\cos x| + C$

• $\int \cot x \, dx = \ln|\sin x| + C$

• $\int \sec x \, dx = \ln|\sec x + \tan x| + C$

• $\int \csc x \, dx = \ln|\csc x - \cot x| + C$

3. 重要高阶/根式积分

• 平方和/差类型 (生成反三角函数):

  • $\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin \frac{x}{a} + C$

  • $\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a} \arctan \frac{x}{a} + C$

  • $\int \frac{1}{a^2-x^2} \, dx = \frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right| + C$

• “长对数”类型:

  • $\int \frac{1}{\sqrt{x^2+a^2}} \, dx = \ln(x + \sqrt{x^2+a^2}) + C$

  • $\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \ln|x + \sqrt{x^2-a^2}| + C$

• 复杂根式 (分部积分推导结果):

  • $\int \sqrt{x^2+a^2} \, dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln(x+\sqrt{x^2+a^2}) + C$
  • $\int \sqrt{x^2-a^2} \, dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}| + C$
  • $\int \sqrt{a^2-x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a} + C$

• 特殊技巧积分:

  • $\int \frac{1}{1+e^x} \, dx = x - \ln(1+e^x) + C$
  • $\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C$
  • $\int \csc^3 x \, dx = -\frac{1}{2}\csc x \cot x + \frac{1}{2}\ln|\csc x - \cot x| + C$

三、 三角函数公式

1. 基础恒等式

• $\sin^2 x + \cos^2 x = 1$

• $\tan^2 x + 1 = \sec^2 x$

• $\cot^2 x + 1 = \csc^2 x$

• $\arcsin x + \arccos x = \frac{\pi}{2}$

2. 倍角与半角公式 (降幂扩角)

• 二倍角:

  • $\sin 2x = 2\sin x \cos x$

  • $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$

• 降幂公式:

  • $\sin^2 x = \frac{1 - \cos 2x}{2}$

  • $\cos^2 x = \frac{1 + \cos 2x}{2}$

3. 和差化积与积化和差

和差化积:

• $\sin \alpha + \sin \beta = 2\sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$

• $\sin \alpha - \sin \beta = 2\cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

• $\cos \alpha + \cos \beta = 2\cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$

• $\cos \alpha - \cos \beta = -2\sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

积化和差:

• $\sin \alpha \sin \beta = -\frac{1}{2}[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$

• $\sin \alpha \cos \beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]$

• $\cos \alpha \sin \beta = \frac{1}{2}[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$

• $\cos \alpha \cos \beta = \frac{1}{2}[\cos(\alpha+\beta) + \cos(\alpha-\beta)]$

四、 麦克劳林公式

1. $\sin x = x - \frac{1}{6}x^3 + o(x^3)$

2. $\arcsin x = x + \frac{1}{6}x^3 + o(x^3)$

3. $\tan x = x + \frac{1}{3}x^3 + o(x^3)$

4. $\arctan x = x - \frac{1}{3}x^3 + o(x^3)$

5. $e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + o(x^3)$

6. $\cos x = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + o(x^4)$

7. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + o(x^3)$

8. $(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2}x^2 + o(x^2)$

五、 反三角函数图像

• $y = \arcsin x$: 定义域 $[-1, 1]$，值域 $[-\frac{\pi}{2}, \frac{\pi}{2}]$，单调递增，奇函数

• $y = \arccos x$: 定义域 $[-1, 1]$，值域 $[0, \pi]$，单调递减，非奇非偶

• $y = \arctan x$: 定义域 $(-\infty, +\infty)$，值域 $(-\frac{\pi}{2}, \frac{\pi}{2})$，单调递增，奇函数

• $y = arccot x$: 定义域 $(-\infty, +\infty)$，值域 $(0, \pi)$，单调递减。