【默写】公式默写自测

一、等价无穷小 (第一讲)

一口气默写十七个常用等价无穷小。

1. 🐶 ~
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   6. ____
2. \(a\) 相关
   1. ____
   2. ____
3. \(\frac{1}{2} 🐶^{2}\)
   1. ____
   2. ____
4. \(e\) 相关
   1. ____
   2. ____
5. 三角函数相关
   1. ____
   2. ____ ~
      1. ____
      2. ____
   3. ____ ~
      1. ____
      2. ____

答案

1. \(🐶\) ~
   1. \(\sin 🐶\)
   2. \(\tan 🐶\)
   3. \(\arcsin 🐶\)
   4. \(\arctan 🐶\)
   5. \(\ln(1+🐶)\)
   6. \(e^🐶-1\)
2. \(a\) 相关
   1. \(🐶\ln a\) ~ \(a^🐶-1\)
   2. \(a🐶\) ~ \((1+🐶)^a-1\)
3. \(\frac{1}{2} 🐶^{2}\) ~
   1. \(1-\cos🐶\)
   2. \(🐶-\ln(1+🐶)\)
4. \(e\) 相关
   1. \(-\frac{e}{2}🐶\) ~ \((1+🐶)^{\frac{1}{🐶}}-e\)
   2. \(e\) ~ \((1+🐶)^{\frac{1}{🐶}}\)
5. 三角函数相关
   1. \(\frac{1}{2}🐶^{3}\) ~ \(\tan🐶-\sin🐶\)
   2. \(\frac{1}{6}🐶^{3}\) ~
      1. \(🐶-\sin🐶\)
      2. \(\arcsin🐶-🐶\)
   3. \(\frac{1}{3}🐶^{3}\) ~
      1. \(\tan🐶-🐶\)
      2. \(🐶-\arctan 🐶\)

二、泰勒展开式

1. ____
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2. _ = _
3. _ = _
4. _ = _
   1. _ = _
   2. _ = _

答案

1. 三角函数
   1. \(\sin x\) = \(x - \frac{x^3}{3!}+O(x^3)\)
   2. \(\cos x\) = \(1 - \frac{x^{2}}{2!}+\frac{x^4}{4!}+O(x^4)\)
   3. \(\tan x\) = \(x+\frac{x^3}{3}+O(x^3)\)
2. 反三角函数
   1. \(\arctan x\) = \(x-\frac{x^3}{3}+O(x^3)\)
   2. \(\arcsin x\) = \(x+\frac{x^3}{3!}+O(x^3)\)
3. 对数
   1. \(\ln(1+x)\) = \(x-\frac{x^{2}}{2}+\frac{x^3}{3}+O(x^3)\)
4. 指数
   1. \(e^x\) = \(1+x+\frac{x^{2}}{2!}+\frac{x^3}{3!}+O(x^3)\)
   2. \((1+x)^a\) = \(1+ax+\frac{a(a-1)}{2}x^{2}+O(x^{2})\)
      1. \(\frac{1}{1-x}=1+x+x^{2}+\dots+x^n+o(x^n)\)
      2. \(\frac{1}{1+x}\) = \(1-x+x^{2}-x^{3}+\dots+(-1)^nx^n+O(x^n), \quad -1<x<1\).

三、基本求导公式

1. ____
   1. _ = _
2. ____
   1. _ = _
   2. _ = _
3. ____
   1. _ = _
   2. _ = _
4. ____
   1. _ = _
   2. _ = _
   3. _ = _
   4. _ = _
   5. _ = _
5. ____
   1. _ = _
   2. _ = _
   3. _ = _
   4. _ = _
6. ____
   1. _ = _
   2. _ = _

答案

1. 幂函数
   1. \((x^a)'\) = \(ax^{a-1}\)
2. 指数函数
   1. \((a^x)'\) = \(a^x\ln a\quad(a>0,a\neq 1)\)
   2. \((e^x)'\) = \(e^x\)
3. 对数函数
   1. \((\log a^x)'\) = \(\frac{1}{x\ln a}\quad(a>0,a\neq 1)\)
   2. \((\ln|x|)'\) = \(\frac{1}{x}\)
4. 三角函数
   1. \((\sin x)'\) = \(\cos x\)
   2. \((\cos x)'\) = \(-\sin x\)
   3. \((\tan x)'\) = \(\sec ^{2} x\)
   4. \((\cot x)'\) = \(-\csc ^{2} x\)
   5. \((\sec x)'\) = \(\sec x\tan x\)
   6. \((\csc x)'\) = \(-\csc x\cot x\)
5. 反三角函数
   1. \((\arcsin x)'\) = \(\frac{1}{\sqrt{ 1-x^{2} }}\)
   2. \((\arccos x)'\) = \(- \frac{1}{\sqrt{ 1-x^{2} }}\)
   3. \((\arctan x)'\) = \(\frac{1}{1+x^{2}}\)
   4. \((arccot x)'\) = \(-\frac{1}{1+x^{2}}\)
6. 补充
   1. \([\ln(x + \sqrt{ x^{2}+1 })]'\) = \(\frac{1}{\sqrt{ x^{2}+1 }}\)
   2. \([\ln(x+\sqrt{ x^{2}-1 })]'\) = \(\frac{1}{\sqrt{ x^{2}-1 }}\)

四、基本积分公式

1. ____
   1. _ = _
      1. _ = _
      2. _ = _
      3. _ = _
2. ____
   1. _ = _
   2. _ = _
3. ____
   1. ____
      1. _ = _
      2. _ = _
   2. ____
      1. _ = _
      2. _ = _
   3. ____
      1. _ = _
      2. _ = _
   4. ____
      1. _ = _
      2. _ = _
   5. ____
      1. _ = _
      2. _ = _
4. ____
   1. _ = _
   2. _ = _
5. ____
   1. _ = _

6. ____

   1. _ = _
   2. _ = _

7. ____

   1. _ = _
   2. _ = _

8. ____

   1. _ = _
9. ____
   1. _ = _
   2. _ = _
10. ____
    1. _ = _
    2. _ = _

答案

1. 幂函数
   1. \(\int x^k dx\) = \(\frac{1}{k+1}x^{k+1} + C,k\neq-1;\)
      1. \(\int \frac{1}{x^{2}}dx\) = \(-\frac{1}{x}+C\)
      2. \(\int \frac{1}{\sqrt{ x }}dx\) = \(2 \sqrt{ x }+ C\)
      3. \(\int \frac{1}{x} dx\) = \(\ln|x| + C\)
2. 指数函数
   1. \(\int e^x dx\) = \(e^x + C\)
   2. \(\int a^x dx\) = \(\frac{a^x}{\ln a} + C, a>0且a\neq 1\)
3. 三角函数
   1. 正余弦
      1. \(\int \sin x dx\) = \(-\cos x + C\)
      2. \(\int \cos xdx\) = \(\sin x+C\)
   2. 正余切
      1. \(\int \tan xdx\) = \(-\ln|\cos x| + C\)
      2. \(\int \cot xdx\) = \(\ln|\sin x|+C\)
   3. 正余割
      1. \(\int \frac{dx}{\cos x}\) = \(\int \sec xdx\) = \(\ln|\sec x+\tan x|+C\)
      2. \(\int \frac{dx}{\sin x}\) = \(\int \csc xdx\) = \(\ln|\csc x-\cot x|+C\)
   4. 正割余割的平方
      1. \(\int \sec ^{2}xdx\) = \(\tan x+C\)
      2. \(\int \csc ^{2}xdx\) = \(-\cot x+C\)
   5. 割切
      1. \(\int\sec x \tan xdx\) = \(\sec x+C\)
      2. \(\int \csc x\cot xdx\) = \(-\csc x+C\)
4. （分母无根号）平方和 \(\begin{cases}\int \frac{1}{1+x^{2}}dx = \arctan x + C, \\\int \frac{1}{a^{2}+x^{2}}dx = \frac{1}{a} \arctan \frac{x}{a} + C(a>0) \\\end{cases}\)
5. （分母无根号）平方差 \(\int \frac{1}{x^{2}-a^{2}}dx = \frac{1}{2a} \ln \left|\frac{{x-a}}{x+a}\right| + C\quad \left( \int \frac{1}{a^{2}-x^{2}}dx= \frac{1}{2a} \ln\left| \frac{{x+a}}{x-a} \right| + C \right).\)
6. （分母根号式）数字开头方差 \(\begin{cases}\int \frac{1}{\sqrt{ 1-x^{2} }} dx = \arcsin x + C \\\int \frac{1}{\sqrt{ a^{2}-x^{2} }} dx = \arcsin \frac{x}{a} + C(>0) \\\end{cases}\)

7. （分母根号式）数字结尾含平方 \(\begin{cases}\int \frac{1}{\sqrt{ x^{2}+a^{2} }}dx = \ln(x+\sqrt{ x^{2}+a^{2} }) +C(常见a=1)\\\int \frac{1}{\sqrt{ x^{2}-a^{2} }}dx = \ln|x+\sqrt{ x^{2}-a^{2} }|+C(|x|>a).\end{cases}\)

8. 非分式数字开头方差 \(\int \sqrt{ a^{2}-x^{2} }dx = \frac{a^{2}}{2}\arcsin \frac{x}{a} + \frac{x}{2}\sqrt{ a^{2}-x^{2} } + C(a>|x|\geq 0).\)

9. 正余弦方
   1. \(\int \sin ^{2}xdx\) = \(\frac{x}{2} - \frac {{\sin 2x}} {4} + C \quad \left( \sin ^{2}x= \frac{{1-\cos 2x}}{2} \right)\)
   2. \(\int \cos ^{2}xdx\) = \(\frac{x}{2} + \frac {{\sin 2x}} {4} + C \quad \left( \cos ^{2}x = \frac{{1 +\cos 2x}}{2} \right)\)
10. 正余切方
    1. \(\int \tan ^{2}xdx\) = \(\tan x-x+C\quad(\tan ^{2}x=\sec ^{2}x-1)\)
    2. \(\int \cot ^{2}xdx\) = \(-\cot x-x+C\quad(\cot ^{2}x=\csc ^{2}x-1)\)

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