由于高数公式比较多，一次也记不住，需要经常翻阅书籍，因此，对高数中常用的公式进行了整理，因为运算中经常出现中学的公式，所以也将中学公式进行了整理。中学公式乘法立方和公式：\mathop{{a}}\nolimits^{{3}}+\mathop{{b}}\nolimits^{{3}}={ \left( {a+b} \right) }{ \left( {\mathop{{a}}\nolimits^{{2}}-ab+\mathop{{b}}\nolimits^{{2}}} \right) } 立方差公式：\mathop{{a}}\nolimits^{{3}}-\mathop{{b}}\nolimits^{{3}}={ \left( {a-b} \right) }{ \left( {\mathop{{a}}\nolimits^{{2}}+ab+\mathop{{b}}\nolimits^{{2}}} \right) } 完全立方公式：\begin{array}{*{20}{l}} {\mathop{{ \left( {a+b} \right) }}\nolimits^{{3}}=\mathop{{a}}\nolimits^{{3}}+3\mathop{{a}}\nolimits^{{2}}b+3a\mathop{{b}}\nolimits^{{2}}+\mathop{{b}}\nolimits^{{3}}}\\ {\mathop{{ \left( {a-b} \right) }}\nolimits^{{3}}=\mathop{{a}}\nolimits^{{3}}-3\mathop{{a}}\nolimits^{{2}}b+3a\mathop{{b}}\nolimits^{{2}}-\mathop{{b}}\nolimits^{{3}}} \end{array} 十字相乘法：\left( {x+a} \left) { \left( {x+b} \right) }=\mathop{{x}}\nolimits^{{2}}+{ \left( {a+b} \right) }x+ab\right. \right. 一元二次方程：\begin{array}{*{20}{l}} {a\mathop{{x}}\nolimits^{{2}}+bx+c=0}\\ { \Delta =\mathop{{b}}\nolimits^{{2}}-4ac}\\
{\mathop{{x}}\nolimits_{{1,2}}=\frac{{-b \pm \sqrt{{\mathop{{b}}\nolimits^{{2}}-4ac}}}}{{2a}}}\\ {\mathop{{x}}\nolimits_{{1}}+\mathop{{x}}\nolimits_{{2}}=-\frac{{b}}{{a}}}\\ {\mathop{{x}}\nolimits_{{1}}\mathop{{x}}\nolimits_{{2}}=\frac{{c}}{{a}}} \end{array} 裂项法\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}\\ \frac{1}{n(n+k)}=\frac{1}{k}(\frac{1}{n}-\frac{1}{n+k})\\
\frac{1}{\sqrt n +\sqrt{(n+1)}}=\sqrt{n+1}-\sqrt{n}\\ \frac{1}{\sqrt n +\sqrt{(n+k)}}=\frac{1}{k}(\sqrt{n+k}-\sqrt{n})\\
常用不等式三角不等式：{ \left
a+b \left
\le
\left
a \left
+ \left
b \right
\right. \right. \right. \right. }
\left
a-b \left
\le
\left
a \left
+ \left
b \right
\right. \right. \right. \right.
\left
a-b \left
\ge
\left
a \left
- \left
b \right
\right. \right. \right. \right.
\left
a \left
- \left
b \left
\le
\left
a+b \left
\le
\left
a \left
+ \left
b \right
\right. \right. \right. \right. \right. \right. \right. \right. 正弦余弦不等式：\begin{array}{*{20}{l}} {\text{当}x \in
\left( 0,\frac{{ \pi }}{{2}} \left) ,\right. \right. }\\ {\text{则}\text{s}\text{i}\text{n}x < x < \text{t}\text{a}\text{n}x} \end{array} \begin{array}{*{20}{l}} {\text{当}x \in
\left( 0,\frac{{ \pi }}{{2}} \left) ,\right. \right. }\\ {\text{则}1 < \text{s}\text{i}\text{n}x+\text{c}\text{o}\text{s}x \le \sqrt{{2}}} \end{array} 均值不等式：\begin{array}{*{20}{l}} {\text{当}x \in
\left( 0,\frac{{ \pi }}{{2}} \left) ,\right. \right. }\\ {\text{则}1 < \text{s}\text{i}\text{n}x+\text{c}\text{o}\text{s}x \le \sqrt{{2}}} \end{array} 三角函数倍角公式 \begin{array}{*{20}{l}} { \text{sin} 2 \alpha =2 \text{sin}
\alpha
\text{cos}
\alpha }\\ { \text{cos} 2 \alpha \begin{array}{*{20}{l}} {={\mathop{{ \text{cos} }}\nolimits^{{2}} \alpha -}\mathop{{ \text{sin} }}\nolimits^{{2}} \alpha }\\ {=2\mathop{{ \text{cos} }}\nolimits^{{2}} \alpha -1}\\ {=1-2\mathop{{ \text{sin} }}\nolimits^{{2}} \alpha } \end{array}}\\ { \text{tan} 2 \alpha =\frac{{2 \text{tan}
\alpha }}{{1-\mathop{{ \text{tan} }}\nolimits^{{2}} \alpha }}} \end{array}
和差化积公式 \begin{array}{*{20}{l}} { \text{sin}
\alpha + \text{sin}
\beta =2 \text{sin} \frac{{ \alpha + \beta }}{{2}} \text{cos} \frac{{ \alpha - \beta }}{{2}}}\\ { \text{sin}
\alpha - \text{sin}
\beta =2 \text{cos} \frac{{ \alpha + \beta }}{{2}} \text{sin} \frac{{ \alpha - \beta }}{{2}}}\\ { \text{cos}
\alpha + \text{cos}
\beta =2 \text{cos} \frac{{ \alpha + \beta }}{{2}} \text{cos} \frac{{ \alpha - \beta }}{{2}}}\\ { \text{cos}
\alpha - \text{cos}
\beta =-2 \text{sin} \frac{{ \alpha + \beta }}{{2}} \text{sin} \frac{{ \alpha - \beta }}{{2}}} \end{array}
加法公式 \begin{array}{*{20}{l}} { \text{sin} { \left( { \alpha + \beta } \right) }= \text{sin}
\alpha
\text{cos}
\beta + \text{cos}
\alpha
\text{sin}
\beta }\\ { \text{sin} { \left( { \alpha - \beta } \right) }= \text{sin}
\alpha
\text{cos}
\beta - \text{cos}
\alpha
\text{sin}
\beta }\\ { \text{cos} { \left( { \alpha + \beta } \right) }= \text{cos}
\alpha
\text{cos}
\beta - \text{sin}
\alpha
\text{sin}
\beta }\\ { \text{cos} { \left( { \alpha - \beta } \right) }= \text{cos}
\alpha
\text{cos}
\beta + \text{sin}
\alpha
\text{sin}
\beta }\\ { \text{tan} { \left( { \alpha + \beta } \right) }=\frac{{ \text{tan}
\alpha + \text{tan}
\beta }}{{1- \text{tan}
\alpha
\text{tan}
\beta }}}\\ { \text{tan} { \left( { \alpha - \beta } \right) }=\frac{{ \text{tan}
\alpha - \text{tan}
\beta }}{{1- \text{tan}
\alpha
\text{tan}
\beta }}} \end{array}
恒等式 \begin{array}{*{20}{l}} {\mathop{{ \text{sin} }}\nolimits^{{2}} \alpha +\mathop{{ \text{cos} }}\nolimits^{{2}}=1}\\ {\mathop{{ \text{tan} }}\nolimits^{{2}} \alpha +1=\mathop{{ \text{sec} }}\nolimits^{{2}} \alpha }\\ {\mathop{{ \text{cot} }}\nolimits^{{2}} \alpha +1=\mathop{{ \text{csc} }}\nolimits^{{2}} \alpha } \end{array}
极限常用的基本极限函数极限： \lim_{x\to 0}{\frac{\sin x}{x}}=1\\ \\ \lim_{x\to 0}{(1+x)^\frac{1}{x}}=e\\ \lim_{x\to \infty}{(1+\frac{1}{x})^x}=e\\ \\ \lim_{x\to 0}{\frac{a^x-1}{x}}=\ln a\\ \\ \lim_{x\to \infty}{\frac{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+...+b_1x+a_0}}= \begin{cases}
\frac{a_n}{b_m}, &n=m\\
0, &n<m\\
\infty, &n>m \end{cases}
数列极限： \lim_{n\to \infty}{\sqrt[n]n}=1\\ \\ \lim_{n\to \infty}{\sqrt[n]a}=1(a>0)\\
\lim_{x\to \infty}{x^n}= \begin{cases}
0, &|x|<1\\
\infty, &|x|>1\\
1, &x=1\\
\text{不存在},&x=-1 \end{cases} \\ \\ \lim_{x\to \infty}{e^{nx}}= \begin{cases}
0, &x<0\\
+\infty, &x>0\\
1, &x=0 \end{cases}
常用的等价无穷小当 x → 0 时， x\sim \sin x\\ \sim \tan x\\ \sim \arcsin x\\ \sim \arctan x\\ \sim \ln (1+x)\\ \sim e^x-1
(1+x)^a-1 \sim ax\\
a^x-1 \sim x\ln a\\ \\ \boxed{ 若a(x)→0,a(x)b(x)→0,\\ 则 (1+a(x))^{b(x)}-1 \sim a(x)b(x)}
1-\cos x\sim \frac{1}{2}x^2\\ x-\ln (1+x) \sim \frac{1}{2}x^2\\ \\ x-\sin x \sim \frac{1}{6}x^3\\ \arcsin x-x \sim \frac{1}{6}x^3\\ \\ \tan x- x \sim \frac{1}{3}x^3\\ x-\arctan x \sim \frac{1}{3}x^3\\
导数基本公式 \begin{array}{*{20}{l}} { \left( {C} \left)
\prime =0\right. \right. }\\ { \left( {\mathop{{x}}\nolimits^{{ \mu }}} \left)
\prime = \mu \mathop{{x}}\nolimits^{{ \mu -1}}\right. \right. } \end{array}
\begin{array}{*{20}{l}} { \left( { \text{sin} x} \left)
\prime = \text{cos} x\right. \right. }\\ { \left( { \text{cos} x} \left)
\prime =- \text{sin} x\right. \right. }\\ { \left( { \text{tan} x} \left)
\prime =\mathop{{ \text{sec} }}\nolimits^{{2}}x\right. \right. }\\ { \left( { \text{cot} x} \left)
\prime =-\mathop{{ \text{csc} }}\nolimits^{{2}}x\right. \right. }\\ { \left( { \text{sec} x} \left)
\prime = \text{sec} x \text{tan} x\right. \right. }\\ { \left( { \text{csc} x} \left)
\prime =- \text{csc} x{ \text{cot} x}\right. \right. } \end{array}
\begin{array}{*{20}{l}} { \left( {\mathop{{a}}\nolimits^{{x}}} \left)
\prime =\mathop{{a}}\nolimits^{{x}} \text{ln} a\right. \right. }\\ { \left( {\mathop{{e}}\nolimits^{{x}}} \left)
\prime =\mathop{{e}}\nolimits^{{x}}\right. \right. }\\ { \left( {\mathop{{ \text{log} }}\nolimits_{{a}}x} \left)
\prime =\frac{{1}}{{x \text{ln} a}}\right. \right. }\\ { \left( { \text{ln} a} \left)
\prime =\frac{{1}}{{x}}\right. \right. } \end{array}
\begin{array}{*{20}{l}} { \left( { \text{arcsin} x} \left)
\prime =\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}}\right. \right. }\\ { \left( { \text{arccos} x} \left)
\prime =-\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}}\right. \right. }\\ { \left( { \text{arctan} x} \left)
\prime =\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}}\right. \right. }\\ { \left( { \text{arccot} x} \left)
\prime =-\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}}\right. \right. } \end{array}
求导法则有理运算法则：设 u =u(x)，v = v(x) 在 x 处可导，则 \begin{array}{*{20}{l}} { \left( {u \pm v} \left)
\prime ={u \prime } \pm {v \prime }\right. \right. }\\ { \left( {Cu} \left)
\prime =C{u \prime }\right. \right. }\\ { \left( {uv} \left)
\prime ={u \prime }v+u{v \prime }\right. \right. }\\ { \left( {\frac{{u}}{{v}}} \left)
\prime =\frac{{u \prime v-u{v \prime }}}{{\mathop{{v}}\nolimits^{{2}}}},{ \left( {v \neq 0} \right) }\right. \right. } \end{array}
复合函数求导： \begin{array}{*{20}{l}} {y=f{ \left( {u} \right) },u=g{ \left( {x} \right) }}\\ {\frac{{ \text{d} y}}{{ \text{d} x}}=\frac{{ \text{d} y}}{{ \text{d} u}} \cdot \frac{{ \text{d} u}}{{ \text{d} x}}} \end{array}
隐函数求导：设 y = f(x) 是由方程 F(x,y) = 0 所确定的可导函数，为求得 y' ，可在方程 F(x,y) = 0 两边对 x 求导，可得到一个含有 y' 的方程，从中解出 y' 即可 y' 也可由多元函数微分法中的隐函数求导公式\frac{dy}{dx}=-\frac{F'_x}{F'_y}得到
反函数求导：若 y = f(x) 在某区间内可导，且 f'(x) ≠ 0，则其反函数 x = φ(y) 在对应区间内也可导，且 \varphi'(x)=\frac{1}{f'(x)}\\ 即\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}
参数方程求导 设 y=y(x)是由参数方程\begin{cases} x=\varphi(t),\\ &(\alpha \leq t \leq \beta)确定的函数，则\\ y=\psi(t), \end{cases} \\\\ (1)若\varphi(t)和\psi(t)都可导,且\varphi'(t)\not=0,则\\ \frac{dy}{dx}=\frac{\psi'(t)}{\varphi'(t)}\\ \\ (2)若\varphi(t)和\psi(t)二阶可导,且\varphi'(t)\not=0,则\\ \frac{d^2y}{dx^2}=\frac{d}{dt}(\frac{\psi'(t)}{\varphi'(t)})\cdot\frac{1}{\varphi'(t)}\\ \\ =\frac{\psi''(t)\varphi'(t)-\varphi''(t)\psi'(t)}{\varphi'^3(t)}
常用的高阶导数公式： (\sin x)^{(n)}=\sin(x+n·\frac{\pi}{2})\\
(\cos x)^{(n)}=\cos(x+n·\frac{\pi}{2})\\
(u\pm v)^{(n)}=(u)^{(n)}\pm (v)^{(n)}\\
(uv)^{(n)}=\sum_{k=0}^{n}C^k_nu^{(k)}v^{(n-k)}
不定积分基本公式 \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}
\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}\\ \end{array}
\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}
{\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}}
\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}
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}
三角换元含有 a2 - x2 的积分，令 x = a sint含有 a2 + x2 的积分，令 x = a tant含有 x2 - a2 的积分，令 x = a sect三角有理式积分 \int R(\sin x,\cos x)dx
（1）一般方法（万能代换） 令\tan \frac{x}{2}=t \\ \int R(\sin x,\cos x)dx = \int R(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}) \frac{2}{1+t^2}dt
（1）特殊方法（三角变形，换元，分部）
几种常用的换元法 1.若R(-\sin x,\cos x)=-R(\sin x,\cos x),则令u=\cos x,或凑d\cos x \\ 2.若R(\sin x,-\cos x)=-R(\sin x,\cos x),则令u=\sin x,或凑d\sin x \\ 3.若R(-\sin x,-\cos x)=R(\sin x,\cos x),则令u=\tan x,或凑d\tan x
定积分 \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) }
不等式性质 \tag{1}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
\tag{2}\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}
\tag{3}{ \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
}
积分中值定理 \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}
已有公式 \int_{0}^{\frac{\pi}{2}}\sin^nx~dx=\int_{0}^{\frac{\pi}{2}}\cos^nx~dx =\begin{cases} \frac{n-1}{n}·\frac{n-3}{n-2}·...·\frac{1}{2}·\frac{\pi}{2} ,&\text{n为正偶数}\\ \frac{n-1}{n}·\frac{n-3}{n-2}·...·\frac{2}{3} ,&\text{n为大于1的奇数}\\
\end{cases}
\int_{0}^{\pi}xf(\sin x)dx=\frac{\pi}{2}\int_{0}^{\pi}f(\sin x)dx,\text{其中f(x)连续}
反常积分1. 无穷区间上的反常积分：常用结论： \int ^{+\infty}_a \frac{1}{x^p}dx, \begin{cases} p>1,收敛\\ &(a>0)\\ p\leq 1,发散 \end{cases}
2. 无界函数的反常积分：常用结论： \int ^{b}_a \frac{1}{(x-a)^p}dx, \begin{cases} p<1,收敛\\ &\\ p\geq 1,发散 \end{cases} \\ \\ \int ^{b}_a \frac{1}{(b-x)^p}dx, \begin{cases} p<1,收敛\\ &\\ p\geq 1,发散 \end{cases}
定积分应用1. 平面图形的面积：（1）若平面域 D 由曲线 y=f(x)，y=g(x) (f(x)≥g(x)，x=a，x=b (a<b)所围成，则平面域D的面积为： S=\int^b_a[f(x)-g(x)]dx
（2）若平面域 D 由曲线 r=r(θ)，θ=a，θ=β
(a<β)所围成，则其面积为： S=\frac{1}{2}\int^β_\alpha r^2dθ
2. 旋转体体积：若区域 D 由曲线 y=f(x) (f(x)≥0)和直线 x=a，x=b (0≤a<b)及x轴所围成，则（1）区域D绕 x 轴旋转一周所得到的旋转体体积为： V_x=\pi \int^b_a f^2(x)dx
（2）区域D绕 y 轴旋转一周所得到的旋转体体积为： V_y=2\pi \int^b_a x f(x)dx
3. 曲线弧长：（1）C：y = y(x)，a ≤ x ≤ b. s=\int^b_a \sqrt {1+y'^2} dx
(2)~C:\begin{cases} x=x(t),\\ &\qquad\qquad\alpha \leq t \leq \beta\\ y=y(t), \end{cases} \\\\ s=\int^\beta _\alpha \sqrt {r+r'^2} d\theta
4. 旋转体侧面积：曲线 y=f(x) (f(x)≥0)和直线x=a，x=b (0≤a<b)及x轴所围成区域绕x轴旋转，所得旋转体的侧面积为： S=2\pi \int^b_a f(x)\sqrt {1+f'^2(x)} dx
微分方程可分离变量的方程 \begin{array}{*{20}{l}} g(y)dy=f(x)dx\\ \text{求解方法：两端积分}\\ {\frac{{ \text{d} y}}{{ \text{d} x}}=f{ \left( {x} \right) }g{ \left( {y} \right) }}\\ {{}_{ }^{ } \int _{ }^{ }\frac{{ \text{d} y}}{{g{ \left( {y} \right) }}}={}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x+C} \end{array}
一阶线性非齐次微分方程 \begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }y=f{ \left( {x} \right) }}\\ {u={}_{ }^{ } \int _{ }^{ }p{ \left( {x} \right) } \text{d} x} \end{array}}\\ {y=C\mathop{{e}}\nolimits^{{-u}}+\mathop{{e}}\nolimits^{{-u}}{}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) }\mathop{{e}}\nolimits^{{u}} \text{d} x} \end{array}
伯努利方程 \begin{array}{*{20}{l}} {\text{形}\text{如}}\\ {\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }y=f{ \left( {x} \right) }\mathop{{y}}\nolimits^{{n}}{ \left( {n \neq 0,1} \right) }}\\ {\text{称}\text{为}\text{伯}\text{努}\text{利}\text{方}\text{程}\text{,}\text{是}\text{一}\text{种}\text{非}\text{线}\text{性}\text{的}\text{一}\text{阶}\text{微}\text{分}\text{方}\text{程}}\\ {\text{将}\text{伯}\text{努}\text{利}\text{方}\text{程}\text{两}\text{端}\text{除}\text{以}\mathop{{y}}\nolimits^{{n}}\text{,}\text{得}}\\ {\mathop{{y}}\nolimits^{{-n}}\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }\mathop{{y}}\nolimits^{{1-n}}=f{ \left( {x} \right) }}\\ {\text{令}z=\mathop{{y}}\nolimits^{{1-n}}\text{有}}\\ {\frac{{ \text{d} z}}{{ \text{d} x}}={ \left( {1-n} \right) }\mathop{{y}}\nolimits^{{-n}}\frac{{ \text{d} y}}{{ \text{d} x}}}\\ {\frac{{1}}{{1-n}}\frac{{ \text{d} z}}{{ \text{d} x}}+p{ \left( {x} \right) }z=f{ \left( {x} \right) }}\\ {\text{则}\text{原}\text{方}\text{程}\text{化}\text{为}\text{线}\text{性}\text{方}\text{程}\text{进}\text{行}\text{求}\text{解}} \end{array}
全微分方程 \begin{array}{*{20}{l}} {\text{若}\text{方}\text{程}}\\ {P{ \left( {x,y} \right) } \text{d} x+Q{ \left( {x,y} \right) } \text{d} y=0}\\ {\text{的}\text{左}\text{端}\text{是}\text{某}\text{函}\text{数}\text{的}\text{全}\text{微}\text{分}\text{方}\text{程}\text{,}\text{即}}\\ { \text{d} u{ \left( {x,y} \right) }=P{ \left( {x,y} \right) } \text{d} x+Q{ \left( {x,y} \right) } \text{d} y=0}\\ {\text{其}\text{中}}\\ { \left\{ \begin{array}{*{20}{l}} {\frac{{ \partial u}}{{ \partial x}}=P{ \left( {x,y} \right) }}\\ {\frac{{ \partial u}}{{ \partial x}}=Q{ \left( {x,y} \right) }} \end{array}\right. }\\ {\text{则}}\\ {u{ \left( {x,y} \right) }=C}\\ {\text{是}\text{原}\text{方}\text{程}\text{的}\text{通}\text{解}} \end{array}
二阶常系数齐次线性微分方程 \begin{array}{*{20}{l}} {\text{对}\text{二}\text{阶}\text{方}\text{程}}\\ {y '' +p{y \prime }+qy=0}\\ {\text{其}\text{中}p,q\text{为}\text{常}\text{数}}\\ {\text{观}\text{察}\text{其}\text{特}\text{征}\text{方}\text{程}\mathop{{r}}\nolimits^{{2}}+pr+q=0\text{的}\text{根}\mathop{{r}}\nolimits_{{1}}\text{和}\mathop{{r}}\nolimits_{{2}}\text{,}\text{其}\text{通}\text{解}\text{对}\text{照}\text{如}\text{下}}\\
{y={ \left\{ {\begin{array}{*{20}{l}} {\mathop{{C}}\nolimits_{{1}}\mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{1}}x}}+\mathop{{C}}\nolimits_{{2}}\mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{2}}x}}}&{\mathop{{p}}\nolimits^{{2}}-4q > 0 ~(r_1\neq r_2)}\\ { \left( {\mathop{{C}}\nolimits_{{1}}+\mathop{{C}}\nolimits_{{2}}x} \left) \mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{1}}x}}\right. \right. }&{\mathop{{p}}\nolimits^{{2}}-4q=0~(r_1= r_2)}\\ {\mathop{{e}}\nolimits^{{ \alpha x}}{ \left( {\mathop{{C}}\nolimits_{{1}} \text{cos}
\beta x+\mathop{{C}}\nolimits_{{2}} \text{sin}
\beta x} \right) }}&{\mathop{{p}}\nolimits^{{2}}-4q < 0~(r_1=\alpha+i\beta, ~r_2=\alpha-i\beta)} \end{array}}\right. }}\\
{\text{其}\text{中}\mathop{{r}}\nolimits_{{1,2}}= \alpha
\pm i \beta \text{为}\text{特}\text{征}\text{方}\text{程}\text{的}\text{一}\text{对}\text{共}\text{轭}\text{复}\text{根}}\\ { \alpha =-\frac{{p}}{{2}}}\\ { \beta =\frac{{\sqrt{{4q-\mathop{{p}}\nolimits^{{2}}}}}}{{2}}} \end{array}
二阶常系数非齐次线性微分方程 \begin{array}{*{20}{l}} {\text{对}\text{二}\text{阶}\text{方}\text{程}}\\ {y '' +p{y \prime }+qy=f{ \left( {x} \right) }}\\ {\text{先}\text{求}\text{对}\text{应}\text{齐}\text{次}\text{方}\text{程}}\\ {y '' +p{y \prime }+qy=0}\\ {\text{的}\text{通}\text{解}y\text{,}\text{再}\text{根}\text{据}f{ \left( {x} \right) }\text{求}\text{另}\text{一}\text{个}\text{特}\text{解}\mathop{{y}}\nolimits_{{0}}} \end{array}
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