高数常见的数学公式

常见三角函数公式

积化和差公式

$$
\begin{align}
\sin^2 x+\cos^2x & =1\notag \\[7pt]
\sec^2 x-\tan^2x & =1\notag \\[7pt]
\cosh^2x-\sinh^2x & =1\notag
\end{align}
$$

和差化积公式

$$
\begin{align}
\sin\alpha+\sin\beta & =2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\notag \\[7pt]
\sin\alpha-\sin\beta & =2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\notag \\[7pt]
\cos\alpha+\cos\beta & =2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\notag \\[7pt]
\cos\alpha-\cos\beta & =-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\notag \\[7pt]
\tan\alpha+\tan\beta & =\frac{\sin (\alpha+\beta)}{\cos\alpha\cdot\cos \beta}\notag
\end{align}
$$

归一化公式

$$
\begin{align}
\sin^2 x+\cos^2x & =1\notag \\[7pt]
\sec^2 x-\tan^2x & =1\notag \\[7pt]
\cosh^2x-\sinh^2x & =1\notag
\end{align}
$$

倍（半）角公式&降（升）幂公式

$$
\begin{align}
\sin^2x & =\frac{1}{2}(1-\cos 2x)\notag \\[7pt]
\cos^2x & =\frac{1}{2}(1+\cos 2x)\notag \\[7pt]
\tan^2x & =\frac{1-\cos 2x}{1+\cos 2x}\notag \\[7pt]
\sin x & =2\sin\frac{x}{2}\cos\frac{x}{2}\notag \\[7pt]
\cos x & =2\cos^2\frac{x}{2}-1=1-2\sin^2\frac{x}{2}=\cos^2\frac{x}{2}-\sin^2\frac{x}{2}\notag \\[7pt]
\tan x & =\frac{2\tan(x/2)}{1-\tan^2(x/2)}\notag
\end{align}
$$

万能公式

令$ u=\tan\dfrac{x}{2} $则
$$
\begin{align}
\sin x=\frac{2u}{1+u^2}\notag \\[7pt]
\cos x=\frac{1-u^2}{1+u^2}\notag
\end{align}
$$

常用的佩亚诺型余项泰勒公式

$$
\begin{align}
f(x)= & f(x_0)+f’(x_0)(x-x_0)+\frac{f’’(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o[(x-x_0)^n]\notag \\[7pt]
f(x)= & f(x_0)+f’(x_0)(x-x_0)+\frac{f’’(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}\notag
\end{align}
$$

常见的泰勒公式

$$
\begin{align}
\mathrm{e}^{x} & =1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+\cdots+\frac{1}{n!}x^{n}+o(x^{n})\notag \\[7pt]
\ln(x+1) & =x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots+(-1)^{n-1}\frac{1}{n}x^{n}+o(x^{n})\notag
\end{align}
$$

令$ n = 2m $有
$$
\begin{align}
\sin x & =x-\frac{1}{6}x^{3}+\frac{1}{120}x^{5}+\cdots+(-1)^{m-1}\frac{1}{(2m-1)!}x^{2m-1}+o(x^{2m})\notag \\[7pt]
\cos x & =1-\frac{1}{2}x^2+\frac{1}{24}x^4-\cdots+(-1)^m \frac{1}{(2m)!}x^{2m}+o(x^{2m+1})\notag \\[7pt]
\tan x & =x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\cdots+o(x^{2m-1})\notag
\end{align}
$$

常用于近似计算的泰勒公式

$$
\begin{align}
\frac{1}{1-x} & =1+x+x^2+x^3+\cdots+x^n+o(x^n)\notag \\[7pt]
(1+x)^{\alpha} & =\sum_{i=0}^{n}\frac{\prod_{j=0}^{i-1}{(\alpha-j})}{i!}x^n+o(x^n)\notag \\[7pt]
& =1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2+\cdots+o(x^n)\notag \\[7pt]
\alpha^x & =\sum_{i=0}^{n}\frac{\ln^n \alpha}{n!}x^n+o(x^n)\notag \\[7pt]
& =1+x\ln \alpha+\frac{\ln^2 \alpha}{2}x^2+\cdots+\frac{\ln^n \alpha}{n!}x^n+o(x^n)\notag
\end{align}
$$

基本求导公式

$$
\begin{align}
& \left( C\right)’ =0\notag \\[7pt]
& \left( x^{\mu}\right)’ =\mu x^{\mu-1}\notag \\[7pt]
& \left( \sin x\right)’ =\cos x\notag \\[7pt]
& \left( \cos x\right)’ =-\sin x\notag \\[7pt]
& \left( \tan x\right)’ =\sec^2 x\notag \\[7pt]
& \left( \cot x\right)’ =-\csc^2 x\notag \\[7pt]
& \left( \sec x\right)’ =\sec x\cdot\tan x\notag \\[7pt]
& \left( \csc x\right)’ =-\csc x\cdot\tan x\notag \\[7pt]
& \left( a^x\right)’ =a^x\ln a\ (a>0,a\neq1)\notag \\[7pt]
& \left( \log_{a}x\right)’ =\frac{1}{x\cdot\ln a}\ (a>0,a\neq1)\notag \\[7pt]
& \left( \arcsin x\right)’ =\frac{1}{\sqrt{1-x^2}}\notag \\[7pt]
& \left( \arccos x\right)’ =-\frac{1}{\sqrt{1-x^2}}\notag \\[7pt]
& \left( \arctan x\right)’ =\frac{1}{1+x^2}\notag \\[7pt]
& \left( \mathrm{arccot}, x\right)’ =-\frac{1}{1+x^2}\notag
\end{align}
$$

函数图形描述中涉及到的重要公式

常用曲率计算公式

曲率的定义式： $ K=\displaystyle\left|\frac{\mathrm{d}\alpha}{\mathrm{d}s}\right| $

由定义式可以推得

直角坐标系中的曲线$ y=y(x) $有曲率表达式
$$
\begin{align}
K=\frac{\left|y’’\right|}{\left( 1+y^{‘2} \right)^{3/2}}\notag
\end{align}
$$
参数方程表示的曲线$ x=\varphi(t),y=\psi(t) $有曲率表达式
$$
\begin{align}
K=\frac{\left|\varphi’(t)\psi’’(t)-\varphi’’(t)\psi’(t)\right|}{\left[ \varphi^{‘2}(t) +\psi^{‘2}(t) \right]^{3/2}}\notag
\end{align}
$$
极坐标表示的的曲线$ y=y(x) $有曲率表达式
$$
\begin{align}
K=\frac{\left|r^2+2r^{‘2}-r\cdot r’’\right|}\notag{\left(r^2+r^{‘2}\right)^{3/2}}\notag
\end{align}
$$
曲线在对应点$ M(x,y) $的曲率中心$ D(\alpha,\beta) $的坐标为
$$
\begin{align}
\begin{cases}
\alpha = x-\displaystyle\frac{y’(1+y^{‘2})^3}{y^{‘’2}}\notag \\[7pt]
\beta = y+\displaystyle\frac{1+y^{‘2}}{y’’}\notag
\end{cases}
\end{align}
$$

曲线的渐近线

若$ \lim\limits_{ x\rightarrow \infty }f(x)=b $,则称$ y=b $为曲线$ f(x) $的水平渐近线
若$ \lim\limits_{ x\rightarrow x_0 }f(x)=\infty $,则称$ x=x_0 $为曲线$ f(x) $的垂直渐近线
若$ \lim\limits_{ x\rightarrow \infty }[f(x)-(ax+b)]=0 $,其中$ \begin{cases}a=\displaystyle \lim\limits_{x\to \infty}\frac{f(x)}{x} \\[7pt]b=\displaystyle \lim\limits_{x\to \infty}[f(x)-ax]\end{cases} $则称$ y=ax+b $为曲线$ f(x) $的斜渐近线

基本积分公式

$$
\begin{align}
& \int k \,\mathrm{d}x=kx+C \ \mbox{(其中}k\mbox{为常数)}\notag \\[7pt]
& \int x^\mu\,\mathrm{d}x=\frac{x^{\mu+1}}{\mu+1}+C\ (\mu\neq-1)\notag \\[7pt]
& \int \frac{1}{x}\,\mathrm{d}x=\ln|x|+C\notag \\[7pt]
& \int\frac{\mathrm{d}x}{1+x^2}=\arctan x+C\notag \\[7pt]
& \int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin x+C_1=-\arccos x+C_2\notag \\[7pt]
& \int \sin x\,\mathrm{d}x=-\cos x+C\notag \\[7pt]
& \int\cos x \,\mathrm{d}x=\sin x +C\notag \\[7pt]
& \int\tan x\,\mathrm{d}x=-\ln |\cos x|+C\notag \\[7pt]
& \int\cot x\,\mathrm{d}x=\ln |\sin x|+C\notag \\[7pt]
& \int\csc x\,\mathrm{d}x=\int\frac{1}{\sin{x}}\,\mathrm{d}x=\frac{1}{2}
\ln{\left|\frac{1-\cos{x}}{1+\cos{x}}\right|}+C=\ln{\left|\tan{\frac{x}{2}}\right|}+C=\ln{\left|\csc{x}-\cot{x}\right|}+C\notag \\[7pt]
& \int\sec x\,\mathrm{d}x=\int\frac{1}{\cos{x}}\,\mathrm{d}x=\frac{1}{2}
\ln{\left|\frac{1+\sin{x}}{1-\sin{x}}\right|}+C=\ln{\left|\sec{x}+\tan{x}\right|}+C\notag \\[7pt]
& \int\sec^2 x\,\mathrm{d}x=\tan x +C\notag \\[7pt]
& \int \csc^2 x\,\mathrm{d}x=-\cot x +C\notag \\[7pt]
& \int \sec x\cdot\tan x \,\mathrm{d}x=\sec x+C\notag \\[7pt]
& \int\csc x \cdot\cot x \,\mathrm{d}x=-\csc x+C\notag \\[7pt]
& \int \mathrm{e}^x \,\mathrm{d}x=\mathrm{e}^x+C\notag \\[7pt]
& \int a^x\,\mathrm{d}x=\frac{a^x}{\ln a}+C\notag \\[7pt]
& \int \sinh x\,\mathrm{d}x=\cosh x+C\notag \\[7pt]
& \int \cosh x\,\mathrm{d}x=\sinh x+C\notag \\[7pt]
& \int \frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan\frac{x}{a}+C\notag \\[7pt]
& \int \frac{1}{a^2-x^2}\,\mathrm{d}x=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C\notag \\[7pt]
& \int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin\frac{x}{a}+C\notag \\[7pt]
& \int \frac{1}{\sqrt{x^2\pm a^2}}\,\mathrm{d}x=\ln \left|x+\sqrt{x^2\pm a^2}\right|+C\notag
\end{align}
$$

基本积分方法

第一类换元法

三角函数之积的积分

一般地,对于$ \sin^{2k+1}x\cos^n x $或$ \sin^n x \cos^{2k+1}x $ (其中$ k\in\mathbb{N} $) 型函数的积分,总可依次作变换$ u=\cos x $或$ u=\sin x $,从而求得结果
一般地,对于$ \sin^{2k}x\cos^{2l}x $或 (其中$ k,l\in \mathbb{N} $) 型函数的积分,总是利用降幂公式$ \sin^2=\dfrac{1}{2}(1-\cos 2x),\cos^2=\dfrac{1}{2}(1+\cos 2x) $化成$ \cos 2x $的多项式,从而求得结果
一般地,对于$ \tan^{n}x\sec^{2k} x $或$ \tan^{2k-1} x \sec^{n}x $ (其中$ n,k\in\mathbb{N}_{+} $) 型函数的积分,总可依次作变换$ u=\tan x $或$ u=\sec x $,从而求得结果

常见的凑微分类型

$$
\begin{align}
& \int f( ax + b)\rm{d}x = \frac{1}{a}\int f(ax+b)\mathrm{d}(ax + b)\;(a \neq 0)\notag \\[7pt]
& \int f(ax^{m + 1} + b){x^m}{\rm{d}x} = \frac{1}{a(m + 1)}\int f(ax^{m + 1} + b)\rm{d}(ax^{m + 1} + b)\notag \\[7pt]
& \int f\left( \frac{1}{x}\right) \frac{\rm{d}x}{x^2} = - \int f\left( \frac{1}{x}\right) \rm{d}\left( \frac{\rm{1} }{x}\right)\notag \\[7pt]
& \int f(\ln x)\frac{1}{x} \rm{d}x = \int f(\ln x)\rm{d(} \ln x)\notag \\[7pt]
& \int f(\mathrm{e}^x) {\mathrm{e}^x}\rm{d}x = \int f(\mathrm{e}^x )\rm{d(}{\mathrm{e}^x})\notag \\[7pt]
& \int f(\sqrt x )\frac{\rm{d}x}{\sqrt x } = 2\int f(\sqrt x ){\rm{d} }(\sqrt x )\notag \\[7pt]
& \int f(\sin x)\cos x\rm{d}x = \int f(\sin x)\rm{d}\sin x\notag \\[7pt]
& \int f(\cos x)\sin x\rm{d}x = - \int f(\cos x)\rm{d}\cos x\notag \\[7pt]
& \int f(\tan x){\sec }^2 x\rm{d}x = \int f(\tan x)\rm{d}\tan x\notag \\[7pt]
& \int f(\cot x){\csc }^2 x\rm{d}x = - \int f(\cot x)\rm{d}\cot x\notag \\[7pt]
& \int f(\arcsin x)\frac{1}{\sqrt {1 - {x^2} } } \rm{d}x = \int f(\arcsin x)\rm{d}\arcsin x\notag \\[7pt]
& \int f(\arctan x)\frac{1}{1 + {x^2} } \rm{d}x = \int f(\arctan x)\rm{d}\arctan x\notag \\[7pt]
& \int \frac{f’(x)}{f(x)} \rm{d}x = \int \frac{\rm{d}f(x)}{f(x)} = \ln \left| f(x)\right| + C \notag
\end{align}
$$

有理数的积分

部分分式

$$
\begin{align}
\frac{ {P(x)} }{ {Q(x)} } = & \frac{ { {A_1} } }{ { { {(x - a)}^\alpha } } } + \frac{ { {A_2} } }{ { { {(x - a)}^{\alpha - 1} } } } + \cdots + \frac{ { {A_\alpha } } }{ {x - a} } + \notag \\
& \frac{ { {B_1} } }{ { { {(x - b)}^\beta } } } + \frac{ { {B_2} } }{ { { {(x - b)}^{\beta - 1} } } } + \cdots + \frac{ { {B_\beta } } }{ {x - b} } + \notag \\
& \frac{ { {M_1}x + {N_1} } }{ { { {({x^2} + px + q)}^\lambda } } } + \frac{ { {M_2}x + {N_2} } }{ { { {({x^2} + px + q)}^{\lambda - 1} } } } + \cdots + \frac{ { {M_\lambda }x + {N_\lambda } } }{ { {x^2} + px + q} } + \notag \\
& \cdots \notag
\end{align}
$$

三角函数的特殊定积分

$$
\begin{align}
I_n & =\int_0^{\frac{\pi}{2} }\sin^nx,\mathrm{d}x=\int_0^{\frac{\pi}{2} }\cos^nx,\mathrm{d}x\notag \\
I_n & =\frac{n-1}{n}I_{n-2}\notag \\
& =\begin{cases} \notag
\ \dfrac{ {n - 1} }{n} \cdot \dfrac{ {n - 3} }{ {n - 2} } \cdots \dfrac{4}{5} \cdot \dfrac{2}{3}\quad (n\mbox{为大于}1\mbox{的正奇数}),I_1=1 \\[13pt]
\ \dfrac{ {n - 1} }{n} \cdot \dfrac{ {n - 3} }{ {n - 2} } \cdots \dfrac{3}{4} \cdot \dfrac{1}{2} \cdot \dfrac{\pi }{2}\quad (n\mbox{为正偶数}),I_0=\dfrac{\pi}{2}
\end{cases}
\end{align}
$$

多元函数微分

偏导数

偏导数记法

设函数$ z=f(x,y) $在区域$ D $内有偏导数$ \frac{ {\partial z} }{ {\partial x} } = {f_x}(x,y),\quad \frac{ {\partial z} }{ {\partial y} } = {f_y}(x,y) $

他们的偏导数若存在,那么称其偏导数为$ z=f(x,y) $的\md{二阶偏导数}。按照对变量求导次序不同,有如下四个二阶偏导数:
$$
\begin{align}
& \frac{\partial }{ {\partial x} }\left( {\frac{ {\partial z} }{ {\partial x} } } \right) = \frac{ { {\partial ^2}z} }{ {\partial {x^2} } } = {f_{xx} }(x,y) & \frac{\partial }{ {\partial y} }\left( {\frac{ {\partial z} }{ {\partial x} } } \right) = \frac{ { {\partial ^2}z} }{ {\partial x\partial y} } = {f_{xy} }(x,y)\notag \\[7pt]
& \frac{\partial }{ {\partial x} }\left( {\frac{ {\partial z} }{ {\partial y} } } \right) = \frac{ { {\partial ^2}z} }{ {\partial y\partial x} } = {f_{yx} }(x,y) & \frac{\partial }{ {\partial y} }\left( {\frac{ {\partial z} }{ {\partial y} } } \right) = \frac{ { {\partial ^2}z} }{ {\partial {y^2} } } = {f_{yy} }(x,y)\notag
\end{align}
$$