Latex:微积分【大学常用公式】

不定积分1:

$$\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ }k \text{d} x=kx+C}\\
{{}_{ }^{ } \int _{ }^{ }\mathop{{x}}\nolimits^{{ \mu }} \text{d} x=\frac{{\mathop{{x}}\nolimits^{{ \mu +1}}}}{{ \mu +1}}+C,{ \left( { \mu \neq -1} \right) }}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{x}} \text{d} x= \text{ln} { \left| {x} \right| }+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}} \text{d} x= \text{arctan} x+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}} \text{d} x= \text{arcsin} x+C}
\end{array}$$

不定积分2:

\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ } \text{cos} \text{d} x= \text{sin} x+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{sin} x \text{d} x=- \text{cos} x+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{ \text{cos} }}\nolimits^{{2}}x}} \text{d} x={}_{ }^{ } \int _{ }^{ }\mathop{{ \text{sec} }}\nolimits^{{2}}x \text{d} x= \text{tan} x+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{ \text{sin} }}\nolimits^{{2}}x}} \text{d} x={}_{ }^{ } \int _{ }^{ }\mathop{{ \text{csc} }}\nolimits^{{2}}x \text{d} x=- \text{cot} x+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{sec} x \text{tan} x \text{d} x= \text{sec} x+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{csc} x \text{cot} x \text{d} x=- \text{csc} x+C}
\end{array}

不定积分3:

\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ }\mathop{{e}}\nolimits^{{x}} \text{d} x=\mathop{{e}}\nolimits^{{x}}+C}\\
{{}_{ }^{ } \int _{ }^{ }\mathop{{a}}\nolimits^{{x}} \text{d} x=\frac{{\mathop{{a}}\nolimits^{{x}}}}{{ \text{ln} a}}+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{sh} x \text{d} x= \text{ch} x+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{ch} xdx= \text{sh} x+C}
\end{array}

不定积分4:

\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ } \text{tan} x \text{d} x=- \text{ln} { \left| { \text{cos} x} \right| }+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{cot} x \text{d} x= \text{ln} { \left| { \text{sin} x} \right| }+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{sec} x \text{d} x= \text{ln} { \left| { \text{sec} x+ \text{tan} x} \right| }+C}\\
{{}_{ }^{ } \int _{ }^{ } \text{csc} x \text{d} x= \text{ln} { \left| { \text{csc} x- \text{cot} x} \right| }+C}
\end{array}

不定积分5:

\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}} \text{d} x=\frac{{1}}{{a}} \text{arctan} \frac{{x}}{{a}}+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{x}}\nolimits^{{2}}-\mathop{{a}}\nolimits^{{2}}}} \text{d} x=\frac{{1}}{{2a}} \text{ln} { \left| {\frac{{x-a}}{{x+a}}} \right| }+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{a}}\nolimits^{{2}}-\mathop{{x}}\nolimits^{{2}}}}}} \text{d} x= \text{arcsin} \frac{{x}}{{a}}+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}}}} \text{d} x= \text{ln} { \left( {x+\sqrt{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}}} \right) }+C}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{x}}\nolimits^{{2}}-\mathop{{a}}\nolimits^{{2}}}}}} \text{d} x= \text{ln} { \left( {x+\sqrt{{\mathop{{x}}\nolimits^{{2}}\mathop{{a}}\nolimits^{{2}}}}} \right) }+C}
\end{array}

不定积分的性质:

\begin{array}{*{20}{l}}
{{}_{ }^{ } \int _{ }^{ }{ \left[ {f{ \left( {x} \right) }+g{ \left( {x} \right) }} \right] } \text{d} x={}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x+{}_{ }^{ } \int _{ }^{ }g{ \left( {x} \right) } \text{d} x}\\
{{}_{ }^{ } \int _{ }^{ }kf{ \left( {x} \right) } \text{d} x=k{}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x}\\
{{}_{ }^{ } \int _{ }^{ }u \text{d} v=uv-{}_{ }^{ } \int _{ }^{ }v \text{d} u}
\end{array}

第一类曲线积分的性质:

\begin{array}{*{20}{l}}
{L=\mathop{{L}}\nolimits_{{1}}+\mathop{{L}}\nolimits_{{2}} \Rightarrow \mathop{ \int }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s=\mathop{ \int }\nolimits_{{\mathop{{L}}\nolimits_{{1}}}}f{ \left( {x,y} \right) } \text{d} s+\mathop{ \int }\nolimits_{{\mathop{{L}}\nolimits_{{2}}}}f{ \left( {x,y} \right) } \text{d} s}\\
{\mathop{ \iint }\nolimits_{{L}}{ \left[ { \alpha f{ \left( {x,y} \right) }+ \beta f{ \left( {x,y} \right) }} \right] } \text{d} s= \alpha \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s+ \beta \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s}\\
{f{ \left( {x,y} \right) } \le g{ \left( {x,y} \right) } \Rightarrow \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s \le \mathop{ \iint }\nolimits_{{L}}g{ \left( {x,y} \right) } \text{d} s}\\
{ \left| {\mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s} \left| \le \mathop{ \iint }\nolimits_{{L}}{ \left| {f{ \left( {x,y} \right) }} \right| } \text{d} s\right. \right. }
\end{array}

 

定积分1:$a=b \Rightarrow \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=0
$

 

定积分2:$\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=-\mathop{ \int }\nolimits_{{b}}^{{a}}f{ \left( {x} \right) } \text{d} x
$

 

定积分3:$\mathop{ \int }\nolimits_{{a}}^{{b}}{ \left[ {f{ \left( {x} \right) } \pm g{ \left( {x} \right) }} \right] } \text{d} x=\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \pm \mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x
$

 

定积分4:$\mathop{ \int }\nolimits_{{a}}^{{b}}kf{ \left( {x} \right) } \text{d} x=k\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x
$

 

定积分5:$\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=\mathop{ \int }\nolimits_{{a}}^{{c}}f{ \left( {x} \right) } \text{d} x+\mathop{ \int }\nolimits_{{c}}^{{b}}f{ \left( {x} \right) } \text{d} x, \forall c \in { \left( {a,b} \right) }
$

 

定积分6:$\mathop{ \int }\nolimits_{{a}}^{{b}} \text{d} x=b-a
$

 

定积分7:$f{ \left( {x} \right) } \ge 0,x \in { \left[ {a,b} \right] } \Rightarrow \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \ge 0
$

 

定积分8:$f{ \left( {x} \right) } \ge g{ \left( {x} \right) },x \in { \left[ {a,b} \right] } \Rightarrow \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \ge \mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x
$

 

定积分9:$a < b \Rightarrow { \left| {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x} \right| } \le \mathop{ \int }\nolimits_{{a}}^{{b}}{ \left| {f{ \left( {x} \right) } \text{d} x} \right| }
$

 

定积分10:$\begin{array}{*{20}{l}}
{M=\mathop{{f}}\nolimits_{{max}}{ \left( {x} \right) },m=\mathop{{f}}\nolimits_{{min}}{ \left( {x} \right) },x \in { \left[ {a,b} \right] }}\\
{m{ \left( {b-a} \right) } \le \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \le M{ \left( {b-a} \right) }}
\end{array}
$

 

对弧长的曲线积分:

\begin{array}{*{20}{l}}
{\begin{array}{*{20}{l}}
{L}&{\text{光}\text{滑}\text{曲}\text{线}\text{弧}\text{,}\text{积}\text{分}\text{弧}\text{段}}\\
{ \Delta \mathop{{S}}\nolimits_{{i}}}&{\text{第}i\text{个}\text{小}\text{弧}\text{段}}\\
{ \lambda }&{\text{所}\text{有}\text{小}\text{弧}\text{段}\text{长}\text{度}\text{的}\text{最}\text{大}\text{值}}\\
{f{ \left( {x,y} \right) },g{ \left( {x,y} \right) }}&{\text{有}\text{界}\text{函}\text{数}\text{,}\text{被}\text{积}\text{函}\text{数}}\\
{ \text{d} s}&{\text{弧}\text{长}\text{微}\text{元}}\\
{ \alpha , \beta }&{\text{常}\text{数}}
\end{array}}\\
{\mathop{ \int }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s=\mathop{{ \text{lim} }}\limits_{{ \lambda \to 0}}\mathop{ \sum }\limits_{{i=1}}^{{n}}f{ \left( {\mathop{{ \xi }}\nolimits_{{i}},\mathop{{ \eta }}\nolimits_{{i}}} \right) } \Delta \mathop{{s}}\nolimits_{{i}}}\\
{\mathop{ \oint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s}\\
{ \left\{ \begin{array}{*{20}{c}}
{x= \varphi { \left( {t} \right) }}\\
{y= \varphi { \left( {t} \right) }}
\end{array}, \alpha \le t \le \beta \right. }\\
{\mathop{ \int }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s=\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left[ { \varphi { \left( {t} \right) }, \psi { \left( {t} \right) }} \right] }\sqrt{{\mathop{{ \left[ { \varphi \prime { \left( {t} \right) }} \right] }}\nolimits^{{2}}+\mathop{{ \left[ { \psi \prime { \left( {t} \right) }} \right] }}\nolimits^{{2}}}} \text{d} t}
\end{array}

二重积分的定义和计算:

\begin{array}{*{20}{l}}
{\begin{array}{*{20}{l}}
{D}&{\text{有}\text{界}\text{闭}\text{区}\text{域}\text{,}\text{积}\text{分}\text{区}\text{域}}\\
{ \Delta \mathop{{ \sigma }}\nolimits_{{i}}}&{\text{第}i\text{个}\text{小}\text{闭}\text{区}\text{域}}\\
{ \lambda }&{\text{所}\text{有}\text{小}\text{闭}\text{区}\text{域}\text{中}\text{直}\text{径}\text{最}\text{大}\text{的}\text{值}}\\
{f{ \left( {x,y} \right) },g{ \left( {x,y} \right) }}&{\text{定}\text{义}\text{在}D\text{上}\text{的}\text{有}\text{界}\text{函}\text{数}\text{,}\text{被}\text{积}\text{函}\text{数}}\\
{ \sigma }&{D\text{的}\text{面}\text{积}}\\
{ \text{d} \sigma , \text{d} x \text{d} y}&{\text{直}\text{角}\text{坐}\text{标}\text{系}\text{中}\text{的}\text{微}\text{元}\text{面}\text{积}}\\
{ \alpha , \beta }&{\text{常}\text{数}}\\
{m,M}&{\text{在}D\text{上}f{ \left( {x,y} \right) }\text{的}\text{最}\text{小}\text{值}\text{和}\text{最}\text{大}\text{值}}
\end{array}}\\
{\mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma =\mathop{{ \text{lim} }}\limits_{{ \lambda \to 0}}\mathop{ \sum }\limits_{{i=1}}^{{n}}f{ \left( {\mathop{{ \xi }}\nolimits_{{i}},\mathop{{ \eta }}\nolimits_{{i}}} \right) } \Delta \mathop{{ \sigma }}\nolimits_{{i}}}\\
{\mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} x \text{d} y=\mathop{ \int }\nolimits_{{a}}^{{b}}{ \left[ {\mathop{ \int }\nolimits_{{\mathop{{ \varphi }}\nolimits_{{1}}{ \left( {x} \right) }}}^{{\mathop{{ \varphi }}\nolimits_{{2}}{ \left( {x} \right) }}}f{ \left( {x,y} \right) } \text{d} y} \right] } \text{d} x}
\end{array}

二重积分中值定理和换元法:

\begin{array}{*{20}{l}}
{ \exists { \left( { \xi , \eta } \right) } \in D}\\
{\mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma =f{ \left( { \xi , \eta } \right) } \sigma }\\
{\text{设}\text{有}\text{变}\text{换}}\\
{T:x=x{ \left( {u,v} \right) },y=y{ \left( {u,v} \right) },D \to {D \prime }}\\
{\text{且}x{ \left( {u,v} \right) }\text{和}y{ \left( {u,v} \right) }\text{在}{D \prime }\text{上}\text{具}\text{有}\text{一}\text{阶}\text{连}\text{续}\text{偏}\text{导}\text{数}\text{,}\text{并}\text{且}}\\
{J{ \left( {u,v} \right) }=\frac{{ \partial { \left( {x,y} \right) }}}{{ \partial { \left( {u,v} \right) }}} \neq 0}\\
{\mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} x \text{d} y=\mathop{ \iint }\nolimits_{{D}}f{ \left[ {x{ \left( {u,v} \right) },y{ \left( {u,v} \right) }} \right] }{ \left| {J{ \left( {u,v} \right) }} \right| } \text{d} u \text{d} v}
\end{array}

积分中值定理及其推广第一定理:

\begin{array}{*{20}{l}}
{\text{若}\text{函}\text{数}\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{连}\text{续}\text{,}\text{则}}\\
{ \exists \xi \in { \left[ {a,b} \right] }}\\
{\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=f{ \left( { \xi } \left) { \left( {b-a} \right) }\right. \right. }}\\
{\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}\text{且}g{ \left( {x} \right) }\text{在}\text{此}\text{区}\text{间}\text{上}\text{不}\text{变}\text{号}\text{,}\text{则}}\\
{\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( { \xi } \right) }\mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x}
\end{array}

积分中值定理推广第二定理:

\begin{array}{*{20}{l}}
{\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}\text{且}f{ \left( {x} \right) }\text{为}\text{单}\text{调}\text{函}\text{数}\text{,}\text{则}}\\
{ \exists \xi \in { \left[ {a,b} \right] }}\\
{\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {a} \right) }\mathop{ \int }\nolimits_{{a}}^{{ \xi }}g{ \left( {x} \right) } \text{d} x+f{ \left( {b} \right) }\mathop{ \int }\nolimits_{{ \xi }}^{{b}}g{ \left( {x} \right) } \text{d} x}\\
{\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}f{ \left( {x} \right) } \ge 0\text{且}\text{为}\text{单}\text{调}\text{递}\text{减}\text{函}\text{数}\text{,}\text{则}}\\
{ \exists \xi \in { \left[ {a,b} \right] }}\\
{\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {a} \right) }\mathop{ \int }\nolimits_{{a}}^{{ \xi }}g{ \left( {x} \right) } \text{d} x}\\
{\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}f{ \left( {x} \right) } \ge 0\text{且}\text{为}\text{单}\text{调}\text{递}\text{增}\text{函}\text{数}\text{,}\text{则}}\\
{ \exists \xi \in { \left[ {a,b} \right] }}\\
{\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {b} \right) }\mathop{ \int }\nolimits_{{ \xi }}^{{b}}g{ \left( {x} \right) } \text{d} x}
\end{array}

极限:

\begin{array}{*{20}{l}}
{\begin{array}{*{20}{c}}
{C}&{\text{常}\text{数}}\\
{f{ \left( {x} \right) }}&{\text{函}\text{数}}\\
{n}&{\text{正}\text{整}\text{数}}\\
{ \left\{ {\mathop{{x}}\nolimits_{{n}}} \left\} ,{ \left\{ {\mathop{{y}}\nolimits_{{n}}} \right\} }\right. \right. }&{\text{数}\text{列}}
\end{array}}\\
{ \text{lim} { \left[ {Cf{ \left( {x} \right) }} \left] =C{ \left[ { \text{lim} f{ \left( {x} \right) }} \right] }\right. \right. }}\\
{ \text{lim} {\mathop{{ \left[ {f{ \left( {x} \right) }} \right] }}\nolimits^{{n}}}={\mathop{{ \left[ { \text{lim} f{ \left( {x} \right) }} \right] }}\nolimits^{{n}}}}\\
{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\mathop{{x}}\nolimits_{{n}} \pm \mathop{{y}}\nolimits_{{n}}} \right) }=\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}} \pm \mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}\\
{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\mathop{{x}}\nolimits_{{n}} \cdot \mathop{{y}}\nolimits_{{n}}} \right) }=\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}} \cdot \mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}\\
{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}{ \left( {\frac{{\mathop{{x}}\nolimits_{{n}}}}{{\mathop{{y}}\nolimits_{{n}}}}} \right) }=\frac{{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{x}}\nolimits_{{n}}}}{{\mathop{{ \text{lim} }}\limits_{{n \to + \text{ \infty } }}\mathop{{y}}\nolimits_{{n}}}},{ \left( {\mathop{{y}}\nolimits_{{n}} \neq 0,n=1,2,3, \cdots } \right) }}
\end{array}

洛必达法则1:

\begin{array}{*{20}{l}}
{\text{若}\text{函}\text{数}f{ \left( {x} \left) ,g{ \left( {x} \right) }\text{满}\text{足}\text{如}\text{下}\text{条}\text{件}\text{:}\right. \right. }}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}f{ \left( {x} \right) }=0,\mathop{{ \text{lim} }}\limits_{{x \to a}}g{ \left( {x} \right) }=0}\\
{\text{在}\text{点}a\text{的}\text{某}\text{去}\text{邻}\text{域}\text{内}f{ \left( {x} \left) \text{和}g{ \left( {x} \right) }\text{都}\text{可}\text{导}\text{,}\text{且}{g} \prime { \left( {x} \left) \neq 0\right. \right. }\right. \right. }}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A,{ \left( {A\text{为}\text{实}\text{数}} \right) }}\\
{\text{则}\text{有}}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f{ \left( {x} \right) }}}{{g{ \left( {x} \right) }}}=\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A}
\end{array}

洛必达法则2:

\begin{array}{*{20}{l}}
{\text{若}\text{函}\text{数}f{ \left( {x} \left) ,g{ \left( {x} \right) }\text{满}\text{足}\text{如}\text{下}\text{条}\text{件}\text{:}\right. \right. }}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}f{ \left( {x} \right) }= \infty ,\mathop{{ \text{lim} }}\limits_{{x \to a}}g{ \left( {x} \right) }= \infty }\\
{\text{在}\text{点}a\text{的}\text{某}\text{去}\text{邻}\text{域}\text{内}f{ \left( {x} \left) \text{和}g{ \left( {x} \right) }\text{都}\text{可}\text{导}\text{,}\text{且}{g} \prime { \left( {x} \left) \neq 0\right. \right. }\right. \right. }}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A,{ \left( {A\text{为}\text{实}\text{数}\text{,}\text{也}\text{可}\text{为} \pm \infty \text{或} \infty } \right) }}\\
{\text{则}\text{有}}\\
{\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f{ \left( {x} \right) }}}{{g{ \left( {x} \right) }}}=\mathop{{ \text{lim} }}\limits_{{x \to a}}\frac{{f \prime { \left( {x} \right) }}}{{g \prime { \left( {x} \right) }}}=A}
\end{array}

牛顿-莱布尼兹公式:

$\mathop{ \int }\nolimits_{{a}}^{{b}}{F \prime }{ \left( {x} \right) } \text{d} x=F{ \left( {b} \right) }-F{ \left( {a} \right) }$

三重积分的定义和计算:

\begin{array}{*{20}{l}}
{\begin{array}{*{20}{l}}
{ \Omega }&{\text{空}\text{间}\text{有}\text{界}\text{闭}\text{区}\text{域}\text{,}\text{积}\text{分}\text{区}\text{域}}\\
{ \Delta \mathop{{ \upsilon }}\nolimits_{{i}}}&{\text{第}i\text{个}\text{小}\text{闭}\text{区}\text{域}}\\
{f{ \left( {x,y,z} \right) }}&{\text{定}\text{义}\text{在} \Omega \text{上}\text{的}\text{有}\text{界}\text{函}\text{数}\text{,}\text{被}\text{积}\text{函}\text{数}}\\
{ \upsilon }&{ \Omega \text{的}\text{体}\text{积}}\\
{ \text{d} \upsilon , \text{d} x \text{d} y \text{d} z}&{\text{直}\text{角}\text{坐}\text{标}\text{系}\text{中}\text{的}\text{微}\text{元}\text{面}\text{积}}
\end{array}}\\
{\mathop{ \iiint }\nolimits_{{ \Omega }}f{ \left( {x,y,z} \right) } \text{d} \upsilon =\mathop{{ \text{lim} }}\limits_{{ \lambda \to 0}}\mathop{ \sum }\limits_{{i=1}}^{{n}}f{ \left( {\mathop{{ \xi }}\nolimits_{{i}},\mathop{{ \eta }}\nolimits_{{i}},\mathop{{ \zeta }}\nolimits_{{i}}} \right) } \Delta \upsilon }\\
{\mathop{ \iiint }\nolimits_{{ \Omega }}f{ \left( {x,y,z} \right) } \text{d} x \text{d} y \text{d} z=\mathop{ \int }\nolimits_{{a}}^{{b}}{ \left\{ {\mathop{ \int }\nolimits_{{\mathop{{ \varphi }}\nolimits_{{1}}{ \left( {x} \right) }}}^{{\mathop{{ \varphi }}\nolimits_{{1}}{ \left( {x} \right) }}}{ \left[ {\mathop{ \int }\nolimits_{{\mathop{{ {\rm Z} }}\nolimits_{{1}}{ \left( {x,y} \right) }}}^{{\mathop{{ {\rm Z} }}\nolimits_{{2}}{ \left( {x,y} \right) }}}f{ \left( {x,y} \right) } \text{d} z} \right] } \text{d} y} \left\} \text{d} x\right. \right. }}
\end{array}

泰勒公式和拉格朗日余项:

\begin{array}{*{20}{l}}
{f{ \left( {x} \right) }{\begin{array}{*{20}{l}}
{=f{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }+{f \prime }{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }{ \left( {x-\mathop{{x}}\nolimits_{{0}}} \right) }+\frac{{f ” { \left( {\mathop{{x}}\nolimits_{{0}}} \right) }}}{{2!}}\mathop{{ \left( {x-\mathop{{x}}\nolimits_{{0}}} \right) }}\nolimits^{{2}}+ \cdots +\frac{{\mathop{{f}}\nolimits^{{ \left( {n} \right) }}{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }}}{{n!}}\mathop{{ \left( {x-\mathop{{x}}\nolimits_{{0}}} \right) }}\nolimits^{{n}}+\mathop{{R}}\nolimits_{{n}}{ \left( {x} \right) }}\\
{=f{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }+\mathop{ \sum }\limits_{{k=1}}^{{n}}\frac{{\mathop{{f}}\nolimits^{{ \left( {k} \right) }}{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }}}{{k!}}\mathop{{ \left( {x-\mathop{{x}}\nolimits_{{0}}} \right) }}\nolimits^{{k}}+\mathop{{R}}\nolimits_{{n}}{ \left( {x} \right) }}
\end{array}}}\\
{\mathop{{R}}\nolimits_{{n}}{ \left( {x} \right) }=\frac{{\mathop{{f}}\nolimits^{{ \left( {n+1} \right) }}{ \left( { \xi } \right) }}}{{ \left( {n+1} \left) !\right. \right. }}{\mathop{{ \left( {x-\mathop{{x}}\nolimits_{{0}}} \right) }}\nolimits^{{n+1}}},\mathop{{x}}\nolimits_{{0}} \le \xi \le x}
\end{array}

微分法则:

\begin{array}{*{20}{l}}
{ \text{d} { \left( {u \pm v} \left) = \text{d} u \pm \text{d} v\right. \right. }}\\
{ \text{d} { \left( {Cu} \left) =C \text{d} u\right. \right. }}\\
{ \text{d} { \left( {uv} \left) =v \text{d} u+u \text{d} v\right. \right. }}\\
{ \text{d} { \left( {\frac{{u}}{{v}}} \right) }=\frac{{v \text{d} u-u \text{d} v}}{{\mathop{{v}}\nolimits^{{2}}}},{ \left( {v \neq 0} \right) }}
\end{array}

微分形式不变性:

\begin{array}{*{20}{l}}
{y=f{ \left( {u} \right) },u=g{ \left( {x} \right) }}\\
{ \text{d} y={\mathop{{y \prime }}\nolimits_{{x}}} \text{d} x={f \prime }{ \left( {u} \right) }{g \prime }{ \left( {x} \right) } \text{d} x}
\end{array}

重积分性质1:

$\mathop{ \iint }\nolimits_{{D}}{ \left[ { \alpha f{ \left( {x,y} \right) }+ \beta g{ \left( {x,y} \right) }} \right] } \text{d} \sigma = \alpha \mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma + \beta \mathop{ \iint }\nolimits_{{D}}g{ \left( {x,y} \right) } \text{d} \sigma$

重积分性质2:

$D=\mathop{{D}}\nolimits_{{1}}+\mathop{{D}}\nolimits_{{2}} \Rightarrow \mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma =\mathop{ \iint }\nolimits_{{\mathop{{D}}\nolimits_{{1}}}}f{ \left( {x,y} \right) } \text{d} \sigma +\mathop{ \iint }\nolimits_{{\mathop{{D}}\nolimits_{{2}}}}f{ \left( {x,y} \right) } \text{d} \sigma$

重积分性质3:

$f{ \left( {x,y} \right) } \le g{ \left( {x,y} \right) } \Rightarrow \mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma =\mathop{ \iint }\nolimits_{{D}}g{ \left( {x,y} \right) } \text{d} \sigma$

重积分性质4:

$\left| {\mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma } \left| \le \mathop{ \iint }\nolimits_{{D}}{ \left| {f{ \left( {x,y} \right) }} \right| } \text{d} \sigma \right. \right.$

重积分性质5:

$m \le f{ \left( {x,y} \right) } \le M \Rightarrow \mathop{ \iint }\nolimits_{{D}}m \text{d} \sigma \le \mathop{ \iint }\nolimits_{{D}}f{ \left( {x,y} \right) } \text{d} \sigma \le \mathop{ \iint }\nolimits_{{D}}M \text{d} \sigma$

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