CMC公式合集
常用极限
\[\lim\limits_{n\to\infty}\sqrt[n]{n}=1,\lim\limits_{n\to\infty}\sqrt[n]{a}=1(a>0)\]
\[\lim\limits_{n\to\infty}\sqrt[n]{a^ n+b^n}=max(a,b)\]
\[\lim\limits_{n\to0}(\frac{a^ n+b^n+c^n}{3})^{\frac{1}{n}}=abc^{\frac{1}{3}}\]
\[\lim\limits_{n\to\infty}(1+\frac{1}{n})^n=e\]
等价无穷小
\[a^u-1\sim u\ln a\]
\[(1+u)^a-1\sim au\]
三角公式
积化和：
\[ \sin A \cos B = \frac{1}{2} (\sin(A + B) + \sin(A - B)) \]
\[ \cos A \sin B = \frac{1}{2} (\sin(A + B) - \sin(A - B)) \]
\[ \cos A \cos B = \frac{1}{2} (\cos(A + B) + \cos(A - B)) \]
\[ \sin A \sin B = \frac{1}{2} (\cos(A - B) - \cos(A + B)) \]
差化积：
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
\[ \cos A+\cos B=2\cos(2A+B)\cos(2A−B) \]
三角函数间的转换
\[ 1 + tan^2(x)=sec^2(x) \]
\[ 1+cot^2(x)=csc^2(x) \]
\[ \arctan\frac{a-b}{1+ab}=\arctan a-\arctan b(a\geq 0, b\geq 0) \]
\[ \arctan \frac{a+b}{1-ab}=\arctan a+\arctan b(0\leq a\leq 1,0\leq b\leq 1) \]
高阶导数
\[[\ln(1+x)]^{(n)}=(-1)^{n-1}\frac{(n-1)!}{(1+x)^n}\]
\[(\frac{1}{ax+b})^{(n)}=\frac{(-1)^n a^n n!}{(ax+b)^{n+1}}\]
\[(sinx)^{(n)}=sin(x+n\cdot\frac{\pi}{2})\]
\[(cosx)^{(x)}=cos(x+n\cdot\frac{\pi}{2})\]
Taylor展开
\[e^x=\sum\limits_{k=0}^{n}\frac{x^k}{k!}+o(x^n)\]
\[\ln(1+x)=\sum\limits_{k=1}^{n}\frac{(-1)^{k-1}}{k}x^k+o(x^n)\]
\[\sin x=\sum\limits_{k=1}^{n}(-1)^{k-1}\frac{x^{2k-1}}{(2k-1)!}+o(x^{2n})\]
\[\cos x=\sum\limits_{k=0}^{n}(-1)^k\frac{x^{2k}}{(2k)!}+o(x^{2n+1})\]
\[\frac{1}{1-x}=\sum\limits_{k=0}^{n}x^k+o(x^n)\]
曲率
\(y=f(x):\)
\[K=\frac{|y''|}{\sqrt{[1+(y'^2)]^3}}\]
\(x=f(t),y=g(t):\)
\[K=\frac{|f'g''-g'f''|}{(f'^2+g'^2)^{\frac{3}{2}}}\]
\(r=r(\theta):\)
\[K=\frac{|r^2+2r'^2-rr''|}{(r^2+r'^2)^\frac{3}{2}}\]
曲率半径：\(R=\frac{1}{K}\)
积分
| 原函数 | 积分后的函数 |
|---|---|
| \(\sec x\) | \(\ln|\sec x+\tan x|\) |
| \(\csc x\) | \(\ln|\csc x-\cot x|\) |
| \(\tan x\) | \(-\ln |\cos x|\) |
| \(\cot x\) | \(\ln|\sin x|\) |
| \(\sqrt{a^2-x^2}\) | \(\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}arcsin\frac{x}{a}\) |
| \(\sqrt{x^2\pm a^2}\) | \(\frac{x}{2}\sqrt{x^2\pm a^2}+\frac{a^2}{2}\ln|x+\sqrt{x^2\pm a^2}|\) |
| \(\frac{1}{\sqrt{x^2\pm a^2}}\) | \(\ln|x+\sqrt{x^2\pm a^2}|\) |
| \(\frac{1}{(1+x^2)^\frac{3}{2}}\) | \(\frac{x}{\sqrt{1+x^2}}\) |
| \(\int_0^{\frac{\pi}{2}}\sin^{2n}xdx\) | $ $ |
| \(\int_0^{\frac{\pi}{2}}\sin^{2n+1}xdx\) | \(\frac{(2n)!!}{(2n+1)!!}\) |
微分
| 原函数 | 导函数 |
|---|---|
| \(\tan x\) | \(\sec^2 x\) |
| \(\cot x\) | \(-\csc x\) |
| \(\sec x\) | \(\sec x\tan x\) |
| \(\csc x\) | \(-\csc x\cot x\) |
| \(\arcsin x\) | \(\frac{1}{\sqrt{1-x^2}}\) |
| \(\arccos x\) | \(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\arctan x\) | \(\frac{1}{1+x^2}\) |
一阶线性微分方程
\[y'+P(x)y=Q(x)\]
\[y=e^{-\int P(x)dx}[\int Q(x)e^{\int P(x)dx}dx + C]\]
Bernoulii方程：
\[y'+P(x)y=Q(x)y^n\]
左右同除\(y^n\)
二阶常系数微分方程
齐次
\[ y''+ay'+by=0\\ \lambda^2+a\lambda+b=0 \]
实根：两个不同的实根 \(\lambda1\) 和 \(\lambda2\)，则通解为： \[ y= C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x} \] 重根：一个重根 \(\lambda\)，则通解为： \[ y=(C_1 + C_2 x)e^{\lambda x} \] 复根：两个复根 \(\lambda= \alpha \pm \beta i\)，则通解为： \[ y=e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x)) \]
非齐次
\[ y'' +ay ′+by=f(x) \]
如果\(f(x)=P_m(x)e^{\lambda x}\)，则假设 \[ y^*=x^kQ_m(x)e^{\lambda x} \] 如果 \(f(x)=e^{\alpha x}P_m(x)\cos\beta x\)或\(f(x)=e^{\alpha x}P_m(x)\sin\beta x\)，则假设 \[ y^*=x^ke^{\alpha x}[R_m(x)\cos\beta x+T_m(x)\sin \beta x] \]
Maclaurin展开
\[ e^x=1+x+\frac{x^2}{2}+……+\frac{x^n}{n!}+……(-\infty <x < \infty) \]
\[ \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+……+(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}+……(-\infty <x < \infty) \]
\[ \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-……+(-1)^{n-1}\frac{x^n}{n}+……({\color{Tan} -1<x\leq 1{\color{Peach} {\color{Peach} } } } ) \]
\[ \frac{1}{x+1}=\sum\limits_{n=0}^{\infty}(-1)^nx^n({\color{Tan} -1<x<1} ) \]
\[ \frac{1}{\sqrt{1+x}}=1+\sum\limits_{n=1}^{\infty}(-1)^n\frac{(2n-1)!!}{(2n)!!}x^n({\color{Tan} -1<x\leq 1} ) \]
积分的奇偶性+周期性
\[\int_0^\pi xf(sinx)dx=\pi\int_0^{\frac{\pi}{2}}f(sinx)dx=\pi\int_0^{\frac{\pi}{2}}f(cosx)dx\]
\[\int_0^{\pi}f(sinx)dx=2\int_0^{\frac{\pi}{2}}f(sinx)dx=2\int_0^{\frac{\pi}{2}}f(cosx)dx\]
\[\int_0^{\pi}f(cos)dx=\left\{\begin{array}{**lr**}2\int_0^{\frac{\pi}{2}}f(cosx)dx & &f偶\\0 & &f奇\end{array}\right.\]
\[\int_{-a}^{a}f(x)dx=\int_0^a[f(x)+f(-x)]dx\]
若\(f(x)\)以T为周期，则\[\int_a^{a+T}f(x)dx=\int_0^Tf(x)dx\]
