查看“高等数学”的源代码

== 极限 ==
当<math>x\to 0</math>时，常用的等价无穷小

（1）<math>x\sim \sin x\sim\tan x\sim\arcsin x\sim\arctan x\sim\ln_{}{\left ( 1+x \right ) } \sim e^{x} -1</math>

（2）<math>1-\cos x\sim \frac{x^{2}}{2} ,1-\cos ^{a} x\sim\frac{a}{2} x^{2} </math>

（3）<math>\left ( 1+x \right ) ^{a} -1\sim ax</math>

（4）<math>a^{x} -1\sim x\ln_{}{a} </math>
{| class="wikitable"
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!<math>\lim_{\bigtriangleup \to 0} \frac{\sin \bigtriangleup }{\bigtriangleup } =1</math>
!<math>\lim_{\bigtriangleup \to 0} \left ( 1+\bigtriangleup  \right ) ^{\frac{1}{\bigtriangleup } } =e</math>
|}

== 泰勒公式 ==
如果函数<math>f\left ( x \right )</math>在<math>x=x_{0} </math>的领域内具有n+1阶导数则
<math>f\left(x\right) =f\left( x_{0} \right)+f'\left ( x_{0}\right)\left(x-_{0}\right )+\cdots+\frac{f^{\left(n\right)}\left( x_{0}\right )}{n!}\left(x-x_{0}\right )^{n}+R_{n}\left(x\right ) </math>

其中<math>\xi </math>介于<math>x</math>与<math>x_{0} </math>之间，<math>R_{n}\left(x\right )  =\frac{f^{\left(n+1\right)}\left( \xi \right)}{\left(n+1\right)!}\left(x-x_{0}\right)^{n+1}</math>称之为拉格朗日余项，余项<math>R_{n}\left(x\right)</math>也可以表示为<math>R_{n}\left(x\right)=o\left(\left(x-x_{0}\right)^{n}\right )</math>。

（1）当<math>x_{0}=0</math>时，<math>f\left ( x \right ) =f\left ( 0 \right ) +f^{'} \left ( 0 \right ) x+\cdots +\frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x\right)^{n}+R_{n} \left ( x \right ) </math>称为麦克劳林公式。

（2）常用的麦克劳林公式：

<math>\ \ \ \ </math><big>①</big><math>\ \ e^{x} =1+x+\frac{x^{2} }{2!} +\cdots +\frac{x^{n} }{n!} + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>②</big><math>\ \ \sin x=x-\frac{x^{3} }{3!}+\cdots +\frac{\left ( -1 \right )^{n}  }{\left ( 2n+1 \right )! } x^{2n+1} + o \left ( x^{2n+1}  \right ) </math>

<math>\ \ \ \ </math><big>③</big><math>\ \ \cos x=1-\frac{x^{2} }{2!} +\cdots +\frac{\left ( -1 \right )^{n}  }{\left ( 2n \right )! } x^{2n}+ o \left ( x^{2n}  \right )   </math>

<math>\ \ \ \ </math><big>④</big><math>\ \ \frac{1}{1-x} =1+x+x^{2} +\cdots +x^{n} + o \left ( x^{n}  \right )  </math>

<math>\ \ \ \ </math><big>⑤</big><math>\ \ \frac{1}{1+x}=1-x+x^{2} -\cdots +\left ( -1 \right ) ^{n}x^{n}  + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>⑥</big><math>\ \ \ln_{}{\left ( 1+x \right ) } =x-\frac{x^{2} }{x} +\frac{x^{3} }{3} -\cdots +\frac{\left ( -1 \right ) ^{n-1} }{n}x^{n} + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>⑦</big><math>\ \ \left ( 1+x \right )^{a}  =1+ax+\frac{a\left ( a-1 \right ) }{2!}x^{2}+\cdots +\frac{a\left(a-1\right)\cdots \left(a-n+1\right)}{n!}x^{n}  + o \left ( x^{n}  \right )</math>

<math>\ \ \ \ </math><big>⑧</big><math>\ \ \arctan x=x-\frac{x^{3} }{3} +\frac{x^{5} }{5} -\cdots +\frac{\left ( -1 \right )^{n}}{2n+1} x^{2n+1}+ o \left ( x^{2n+1}  \right ) </math>
== 三角函数公式 ==

{| class="wikitable"
<math>a^{2}+b^{2}=c^{2} </math>
|-
!正弦（sin）
|<math>\sin \alpha =\frac{a}{c}</math>
!余割（csc）
|<math>\csc \alpha =\frac{c}{a} </math>
!<math>\sin \alpha \ \csc \alpha =1</math>
|-
!余弦（cos）
|<math>\cos \alpha =\frac{b}{c}</math>
!正割（sec）
|<math>\sec \alpha =\frac{c}{b} </math>
!<math>\cos \alpha \ \sec \alpha =1</math>
|-
!正切（tan）
|<math>\tan \alpha =\frac{a}{b} </math>
!余切（cot）
|<math>\cot \alpha =\frac{b}{a} </math>
!<math>\tan \alpha \ \cot \alpha =1</math>
|}
平方关系：
{| class="wikitable"
|-
|<math>\sin ^{2} \alpha +\cos ^{2} \alpha =1</math>
|<math>1+\cot ^{2} \alpha =csc^{2} \alpha </math>
|<math>\tan ^{2} \alpha +1=\sec ^{2} \alpha </math>
|}
二角和差公式：
{| class="wikitable"
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|<math>\sin \left ( \alpha + \beta  \right ) =\sin \alpha \cos \beta+ \cos \alpha\sin \beta</math>
|colspan="1" rowspan="2"|<math>\tan \left ( \alpha +  \beta  \right )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }  </math>
|-
|<math>\sin \left ( \alpha - \beta  \right ) =\sin \alpha \cos \beta- \cos \alpha\sin \beta</math>
|-
|<math>\cos \left ( \alpha+\beta  \right ) =\cos \alpha \cos \beta -\sin \alpha \sin \beta </math>
|colspan="1" rowspan="2"|<math>\tan \left ( \alpha -  \beta  \right )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }  </math>
|-
|<math>\cos \left ( \alpha-\beta  \right ) =\cos \alpha \cos \beta+\sin \alpha \sin \beta </math>
|}
二倍角公式：
{| class="wikitable"
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|<math>\sin 2\alpha =\sin \alpha \cos \alpha+\sin \alpha\cos \alpha=2\sin \alpha \cos \alpha </math>
|colspan="1" rowspan="2"|<math>\tan2\alpha =\frac{2\tan \alpha }{1-\tan ^{2}\alpha  } </math>
|-
|<math>\cos2 \alpha =\cos ^{2}\alpha  -\sin ^{2}\alpha  =2\cos ^{2}\alpha  -1=1-2\sin ^{2}\alpha</math>
|}
积化和差公式
{| class="wikitable"
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|<math>\sin \alpha \cos\beta =\frac{1}{2}\left[\sin\left(\alpha+\beta \right)+\sin\left(\alpha -\beta\right)\right] </math>
|<math>\cos \alpha \cos \beta =\frac{1}{2} \left [ \cos \left ( \alpha +\beta  \right )+\cos \left ( \alpha -\beta  \right )   \right ] </math>
|-
|<math>\cos \alpha \sin \beta =\frac{1}{2} \left [ \sin \left ( \alpha +\beta  \right ) -\sin \left ( \alpha -\beta  \right )  \right ] </math>
|<math>\sin \alpha \ \sin \beta =\frac{1}{2} \left [ \cos \left ( \alpha +\beta  \right )-\cos \left ( \alpha -\beta  \right )   \right ]  </math>
|}
和差化积公式：
{| class="wikitable"
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|<math>\sin \alpha  +\sin \beta =2\sin \frac{\alpha +\beta }{2} \cos \frac{\alpha -\beta }{2} </math>
|<math>\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2} \cos\frac{\alpha -\beta }{2}  </math>
|-
|<math>\sin \alpha  -\sin \beta =2\cos \frac{\alpha +\beta }{2} \sin \frac{\alpha -\beta }{2}  </math>
|<math>\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2} \sin \frac{\alpha -\beta }{2} </math>
|}
万能公式：
{| class="wikitable"
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|<big><math>\sin \alpha =\frac{2\tan \frac{\alpha }{2} }{1+\tan ^{2}\alpha\frac{\alpha }{2}  }</math>
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|<big><math>\cos \alpha =\frac{1-\tan ^{2}\frac{\alpha }{2}  }{1+\tan ^{2}\frac{\alpha }{2}  } </math>
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|<big><math>\tan \alpha =\frac{2\tan \alpha\frac{\alpha }{2} }{1-\tan ^{2}\frac{\alpha }{2}  } </math>
|}
== 不定积分常用三角函数公式 ==
{| class="wikitable"
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|<math>\left ( \sin x \right ) '=\cos x</math>
|<math>\int \cos x\;dx=\sin x+C</math>
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|<math>\left ( \cos x \right ) '=-\sin  x </math>
|<math>\int \sin x\;dx=-\cos  x+C</math>
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|<math>\left ( \tan x \right )'=\sec^2 x </math>
|<math>\int \sec ^{2} x\;dx=\tan x+C</math>
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|<math>\left ( \sec x \right )'=\sec x\,\tan x </math>
|<math>\int \sec x\tan x\;dx=\sec x+C</math>
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|<math>\left ( \cot x \right )'=-\csc^2 x </math>
|<math>\int \csc  ^{2} x\;dx=-\cot x+C</math>
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|<math>\left ( \csc x \right )'=- \csc x\,\cot x </math>
|<math>\int \csc x\cot x\;dx=-\csc x+C</math>
|}

{| class="wikitable"
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|<math>\int \tan x\;dx=\ln_{}{\left | \sec x+\sec x \right | } +C=\ln_{}{\left | \sec x \right | } +C </math>
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|<math>\int \sec x\;dx=\ln_{}{\left | \sec x+\tan x \right | } +C</math>
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|<math>\int \cot x\;dx=-\ln_{}{\left | \sec x+\sec x \right | }+C =-\ln_{}{\left | \sin x \right | } +C</math>
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|<math>\int \csc x\;dx=-\ln_{}{\left | \csc x+\cot x \right | } +C</math>
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|<math>\int \sec ^{3} x\;dx=\frac{1}{2} \left ( \sec x\tan x+\ln_{}{\left | \sec x+\tan x \right | }  \right ) +C</math>
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|<math>\int \csc ^{3} x\;dx=-\frac{1}{2}\left ( \csc x \cot x +\ln_{}{\left | \csc x+\cot x \right | } \right) +C  </math>
|}

{| class="wikitable"
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|<math>\int \frac{dx}{\sqrt{1-x^{2}}}=\arcsin x+C </math>
|<math>\int \frac{dx}{\sqrt{a^{2}-x^{2} } }= \arcsin \frac{x}{a} +C</math>
|-
|<math>\int \frac{dx}{1+x^{2} } =\arctan x+C</math>
|<math>\int \frac{dx}{a^{2}+ x^{2}} =\frac{1}{a} \arctan \frac{x}{a}+C </math>
|}

{| class="wikitable"
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|<math>\int \frac{dx}{x^{2}-a^{2} } =\frac{1}{2a} \ln_{}{\left |\frac{x-a}{x+a}  \right | } +C</math>
|<math>\int \frac{dx}{\sqrt{x^{2}\pm a^{2} } } =\ln_{}{\left | x+\sqrt{x^{2}\pm a^{2}} \right | } +C</math>
|}

{| class="wikitable"
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|<math>\int \sqrt{a^{2}-x^{2} }\;dx=\frac{a^{2} }{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^{2}-x^{2} }+C  </math>
|}