目录！

目录！等价无穷小 $(x\to0)$ 三角函数函数关系诱导公式导数公式泰勒公式泰勒定理常用的泰勒展开式基本积分积分换元法常用凑积分形式积分表（一）含有 $ax+b$ 的积分（二）含有 $\sqrt{ax+b}$ 的积分（三）含有 $x^2\pm a^2$ 的积分（四）含有 $ax^2+b\ \ (a>0)$ 的积分（五）含有 $ax^2+bx+c\ (a>0)$ 的积分（六）含有 $\sqrt{x^2+a^2}\ (a>0)$ 的积分（七）含有 $\sqrt{x^2-a^2}\ (a>0)$ 的积分（八）含有 $\sqrt{a^2-x^2}\ (a>0)$ 的积分（九）含有 $\sqrt{\pm ax^2+bx+c}\ (a>0)$ 的积分（十）含有 $\sqrt{\pm \frac{x-a}{x-b}}$ 或 $\sqrt{(x-a)(b-a)}$ 的积分（十一）含有三角函数的积分（十二）含有反三角函数的积分（其中a>0）（十三）含有指数函数的积分（十四）含有对数函数的积分（十五）含有双曲函数的积分（十六）定积分牛顿-莱布尼兹公式

$(x\to0)$

$\sin x \sim x\ \ \ \ \ \ \arcsin x\sim x$

$\tan x\sim x\ \ \ \ \ \ \arctan x\sim x$

$e^x-1\sim x\ \ \ \ \ \ a^x-1\sim x\ln a$

$\ln(1+x)\sim x\ \ \ \ \ \ \log_a(1+x)\sim x/\ln a$

$1-\cos x\sim 1/2 ^2$

$x-\ln(1+x)\sim1/2\ x^2$

$(1+x)^a-1\sim ax$

$(1+bx)^a-1\sim abx$

$x-\sin x\sim \arcsin x-x\sim1/6\ x^3$

$\tan x-x\sim x-\arctan x\sim 1/3\ x^3$

$\tan x-\sin x\sim \arcsin x-\arctan x\sim 1/2 \ x^3$

三角函数

函数关系

倒数关系
$\sin{\alpha}\cdot\csc{\alpha} = 1$
$\cos{\alpha}\cdot\sec{\alpha} = 1$
$\tan{\alpha}\cdot \cot{\alpha} = 1$
商数关系
$\tan{\alpha}=\frac{\sin{\alpha}}{\cos{\alpha}}$
$\cot{\alpha} = \frac{\cos{\alpha}}{\sin{\alpha}}$
平方关系
$\sin^2\alpha+\cos^2\alpha=1$
$1+\tan^2\alpha=\sec^2\alpha$
$1+\cot^2 \alpha= \csc^2\alpha$

诱导公式

$f(\frac{k}{2}\pi)=g(\alpha),k\in Z$
奇变偶不变，符号看象限

导数公式

$(c)' = 0$

$(x^\mu)'=\mu x^{\mu-1}$

$(\sin x)'=\cos x$

$(\cos x)'=-\sin x$

$(\tan x)'=\sec^2x$

$(\cot x)'=-\csc^2x$

$(\sec x)'=\sec x\cdot \tan x$

$(\csc x)'=-\csc x\cdot \cot x$

$(a^x)'=a^x\ln a$

$(e^x)'=e^x$

$(\log_ax)'=\frac{1}{x\ln a}$

$(\ln x)'=\frac{1}{x}$

$(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$

$(\arccos x)'=-\frac{1}{\sqrt{1-x^2}}$

$(\arctan x)'=\frac{1}{1+x^2}$

$(\arccot x)'=-\frac{1}{1+x^2}$

泰勒公式

泰勒定理

$f(x)= \displaystyle\sum^{n}_{k =0}\frac{f^k(x_0)}{k!}(x-x_0)+\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_0)^{n+1}$

$f(x) = f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}+R_n(x)$

$R_n(x) = o((x-x_0)^n)$
$R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}$

麦克劳林公式
$f(x)= \displaystyle\sum^{n}_{k =0} \frac{f^{(k)}(0)}{k!}x^k+\frac{f^(\xi)}{(n+1)!}$

常用的泰勒展开式

e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{n}}{n!} + o (x^{n})

\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots + (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!} + o (x^{2 n + 1})

\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots + (- 1)^{n} \frac{x^{2 n}}{(2 n)!} + o (x^{2 n})

\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots + (- 1)^{n} \frac{x^{n + 1}}{n + 1} + o (x^{n + 1})

\frac{1}{1 - x} = 1 + x + x^{2} + \dots + x^{n} + o (x^{n})

(1 + x)^{m} = 1 + m x + \frac{m (m + 1)}{2!} x^{2} + \dots + \frac{m (m - 1) \cdot \dots \cdot (m - n + 1)}{n!} x^{n} + o (x^{n})

基本积分

\int k d x = k x + C (k 为 常 数)

\int x^{μ} d x = \frac{x^{μ + 1}}{μ + 1} + C (μ \neq - 1)

\int \frac{d x}{x} = \ln | x | + C

\int \frac{d x}{1 + x^{2}} = \arctan x + C

\int \frac{d x}{\sqrt{1 - x^{2}}} = \arcsin x + C

\int \cos x d x = \sin x + C

\int \sin x d x = - \cos x + C

\int \frac{d x}{\cos^{2} x} = \int \sec^{2} x d x = \tan x + C

\int \frac{d x}{\sin^{2} x} = \int \csc^{2} x d x = - \cot x + C

\int \sec x \tan x d x = \sec x + C

\int \csc x \cot x = - \csc x + C

\int e^{x} d x = e^{x} + C

\int a^{x} d x = \frac{a^{x}}{\ln a} + C

\int \tan x d x = - \ln | \cos x | + C

\int \cot x d x = \ln | \sin x | + C

\int \sec x d x = \ln | \sec x + \tan x | + C

\int \csc x d x = \ln | \csc x - \cot x | + C

\int \frac{d x}{a^{2} + x^{2}} = \frac{1}{a} \arctan \frac{x}{a} + C

\int \frac{d x}{x^{2} - a^{2}} = \frac{1}{2 a} \ln | \frac{x - a}{x + a} | + C

\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \arcsin \frac{x}{a} + C

\int \frac{d x}{\sqrt{x^{2} + a^{2}}} = \ln (x + \sqrt{x^{2} + a^{2}}) + C

\int \frac{d x}{\sqrt{x^{2} - a^{2}}} = \ln | x + \sqrt{x^{2} - a^{2}} | + C

积分换元法

第一类换元法
$\int f [φ (x)] \cdot φ^{'} (x) = [\int f (u) d u]_{u = φ (x)}$
第二类换元法
$\int f (x) d x = [\int f [ψ (t)] ψ^{'} (t) d t]_{t = ψ^{- 1} (x)}$

常用凑积分形式

$d x = \frac{1}{a} d (a x + b)$
$x^{m} d x = \frac{d (x^{m + 1})}{m + 1} = \frac{d (a x^{m + 1} + b)}{(m + 1) a}$
$\frac{1}{x^{2}} d x = - d \frac{1}{x}$
$\frac{d x}{\sqrt{x}} = 2 d \sqrt{x}$
$\frac{d x}{\sqrt{1 - x^{2}}} = d \arcsin x$
$\frac{d x}{1 - x^{2}} = d \arctan x$
$\frac{d x}{\cos^{2} x} = d \tan x$
$e^{x} d x = d e^{x}$
$\frac{d x}{x} = d \ln x$
$\int \frac{f^{'} (x)}{f (x)} d x = \int \frac{d f (x)}{f (x)} = \ln f (x) + C$

积分表

$ax+b$ 的积分

$\int \frac{d x}{a x + b} = \frac{1}{a} \ln | a x + b | + C$
$\int (a x + b)^{μ} d x = \frac{1}{a (μ + 1)} (a x + b)^{μ + 1} + C (μ \neq 1)$
$\int \frac{x}{a x + b} d x = \frac{1}{a^{2}} (a x + b - b \ln | a x + b |) + C$
$\int \frac{x^{2}}{a x + b} d x = \frac{1}{a^{3}} [\frac{1}{2} (a x + b)^{2} - 2 b (a x + b) + b^{2} \ln | a x + b |] + C$
$\int \frac{d x}{x (a x + b)} = - \frac{1}{b} \ln | \frac{a x + b}{x} | + C$
$\int \frac{d x}{x^{2} (a x + b)} = - \frac{1}{b x} + \frac{a}{b^{2}} \ln | \frac{a x + b}{x} | + C$
$\int \frac{x}{(a x + b)^{2}} d x = \frac{1}{a^{2}} (\ln | a x + b | + \frac{b}{a x + b}) + C$
$\int \frac{x^{2}}{(a x + b)^{2}} d x = \frac{1}{a^{3}} (a x + b - b \ln | a x + b | - \frac{b^{2}}{a x + b})$
$\int \frac{d x}{x (a x + b)^{2}} = \frac{1}{b (a x + b)} - \frac{1}{b^{2}} \ln | \frac{a x + b}{x} | + C$

$\sqrt{ax+b}$ 的积分

$\int \sqrt{a x + b} d x = \frac{2}{3 a} \sqrt{(a x + b)^{3}} + C$
$\int x \sqrt{a x + b} d x = \frac{2}{15 a^{2}} (3 a x - 2 b) \sqrt{(a x + b)^{3}} + C$
$\int x^{2} \sqrt{a x + b} d x = \frac{2}{105 a^{3}} (15 a^{2} x^{2} - 12 a b x + 8 b^{2}) \sqrt{(a x + b)^{3}} + C$
$\int \frac{x}{\sqrt{a x + b}} d x = \frac{2}{3 a^{2}} (a x - 2 b) \sqrt{a x + b} + C$
$\int \frac{x^{2}}{\sqrt{a x + b}} d x = \frac{2}{15 a^{3}} (3 a^{2} x^{2} - 4 a b x + 8 b^{2}) \sqrt{a x + b} + C$
$\int \frac{d x}{x \sqrt{a x + b}} = {\begin{cases} \frac{1}{\sqrt{b}} \ln | \frac{\sqrt{a x + b} - \sqrt{b}}{\sqrt{a x + b} + \sqrt{b}} | + C (b > 0), \\ \frac{2}{- \sqrt{- b}} \arctan \sqrt{\frac{a x + b}{- b}} + C (b < 0) . \end{cases}$
$\int \frac{d x}{x^{2} \sqrt{a x + b}} = - \frac{\sqrt{a x + b}}{b x} - \frac{a}{2 b} \int \frac{d x}{x \sqrt{a x + b}}$
$\int \frac{\sqrt{a x + b}}{x} d x = 2 \sqrt{a x + b} + b \int \frac{d x}{x \sqrt{a x + b}}$
$\int \frac{\sqrt{a x + b}}{x^{2}} d x = - \frac{\sqrt{a x + b}}{x} + \frac{a}{2} \int \frac{d x}{x \sqrt{a x + b}}$

$x^2\pm a^2$ 的积分

$\int \frac{d x}{x^{2} + a^{2}} = \frac{1}{a} \arctan \frac{x}{a} + C$
$\int \frac{d x}{(x^{2} + a^{2})^{n}} = \frac{x}{2 (n - 1) a^{2} (x^{2} + a^{2})^{n - 1}} + \frac{2 n - 3}{2 (n - 1) a^{2}} \int \frac{d x}{(x^{2} + a^{2})^{n - 1}}$
$\int \frac{d x}{x^{2} - a^{2}} = \frac{1}{2 a} \ln | \frac{x - a}{x + a} | + C$

目录！

等价无穷小(x→0)(x\to0)