基本积分表#

• $\int k \, dx = kx + C$
• $\int x^\mu \, dx = \frac{x^{\mu+1}}{\mu+1} + C$ ($\mu \neq -1$)
• $\int \frac{1}{x} \, dx = \ln|x| + C$
• $\int a^x \, dx = \frac{a^x}{\ln a} + C$
• $\int e^x \, dx = e^x + C$
• $\int \sin x \, dx = -\cos x + C$
• $\int \cos x \, dx = \sin x + C$
• $\int \sec^2 x \, dx = \tan x + C$
• $\int \csc^2 x \, dx = -\cot x + C$
• $\int \sec x \tan x \, dx = \sec x + C$
• $\int \csc x \cot x \, dx = -\csc x + C$
• $\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C$
• $\int \frac{1}{1+x^2} \, dx = \arctan x + C$
• $\int \tan x \, dx = -\ln|\cos x| + C$
• $\int \cot x \, dx = \ln|\sin x| + C$

重要补充公式#

• $\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a} \arctan \frac{x}{a} + C$
• $\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin \frac{x}{a} + C$
• $\int \frac{1}{x^2-a^2} \, dx = \frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right| + C$
• $\int \frac{1}{\sqrt{x^2 \pm a^2}} \, dx = \ln |x + \sqrt{x^2 \pm a^2}| + C$

基本初等函数导数#

• $(C)' = 0$
• $(x^\mu)' = \mu x^{\mu-1}$
• $(a^x)' = a^x \ln a$
• $(e^x)' = e^x$
• $(\log_a x)' = \frac{1}{x \ln a}$
• $(\ln x)' = \frac{1}{x}$
• $(\sin x)' = \cos x$
• $(\cos x)' = -\sin x$
• $(\tan x)' = \sec^2 x$
• $(\cot x)' = -\csc^2 x$
• $(\sec x)' = \sec x \tan x$
• $(\csc x)' = -\csc x \cot x$
• $(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}$
• $(\arccos x)' = -\frac{1}{\sqrt{1-x^2}}$
• $(\arctan x)' = \frac{1}{1+x^2}$
• $(\text{arccot } x)' = -\frac{1}{1+x^2}$

求导法则#

• $(u \pm v)' = u' \pm v'$
• $(uv)' = u'v + uv'$
• $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
• $[f(g(x))]' = f'(g(x)) \cdot g'(x)$

定积分基本公式与性质#

• $\int_a^b f(x) dx = F(b) - F(a)$
• $\int_a^b [kf(x) \pm lg(x)] dx = k\int_a^b f(x) dx \pm l\int_a^b g(x) dx$
• $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

定积分计算技巧#

• $\int_a^b f(x) dx = \int_\alpha^\beta f[\varphi(t)] \varphi'(t) dt$
• $\int_a^b u dv = [uv]_a^b - \int_a^b v du$
• 若 $f(x)$ 偶函数，则 $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$
• 若 $f(x)$ 奇函数，则 $\int_{-a}^a f(x) dx = 0$

常用定积分公式#

• $\int_0^{\pi/2} \sin^n x dx = \int_0^{\pi/2} \cos^n x dx = \begin{cases} \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} & n\text{为偶数}\\ \frac{(n-1)!!}{n!!} & n\text{为奇数} \end{cases}$
• $\int_0^{+\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$
• $\int_0^\pi x f(\sin x) dx = \frac{\pi}{2} \int_0^\pi f(\sin x) dx$

反常积分#

• $\int_a^{+\infty} f(x) dx = \lim\limits_{t \to +\infty} \int_a^t f(x) dx$
• $\int_{-\infty}^b f(x) dx = \lim\limits_{t \to -\infty} \int_t^b f(x) dx$
• $\int_a^b f(x) dx = \lim\limits_{t \to a^+} \int_t^b f(x) dx$

积分应用公式#

• 直角坐标面积：$S = \int_a^b |f(x) - g(x)| dx$
• 极坐标面积：$S = \frac{1}{2} \int_\alpha^\beta r^2(\theta) d\theta$
• 绕x轴旋转体体积：$V = \pi \int_a^b [f(x)]^2 dx$
• 绕y轴旋转体体积（柱壳法）：$V = 2\pi \int_a^b x |f(x)| dx$
• 直角坐标弧长：$L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$
• 参数方程弧长：$L = \int_\alpha^\beta \sqrt{[x'(t)]^2 + [y'(t)]^2} dt$
• 极坐标弧长：$L = \int_\alpha^\beta \sqrt{r^2(\theta) + [r'(\theta)]^2} d\theta$

两个重要极限#

1. $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$
2. $\lim\limits_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

连续性相关#

• 零点定理：若 $f(x)$ 在 $[a,b]$ 上连续，且 $f(a) \cdot f(b) < 0$，则存在 $\xi \in (a,b)$，使 $f(\xi) = 0$
• 介值定理：若 $f(x)$ 在 $[a,b]$ 上连续，则对于任意 $C \in [m, M]$，存在 $\xi \in [a,b]$，使 $f(\xi) = C$

一阶微分方程的基本类型#

1. 可分离变量方程：$\frac{dy}{dx} = f(x)g(y)$
2. 齐次方程：$\frac{dy}{dx} = f\left(\frac{y}{x}\right)$
3. 一阶线性微分方程：$\frac{dy}{dx} + P(x)y = Q(x)$ 通解公式：$y = e^{-\int P(x)dx} \left[ \int Q(x)e^{\int P(x)dx}dx + C \right]$
4. 伯努利方程：$\frac{dy}{dx} + P(x)y = Q(x)y^n \quad (n \neq 0,1)$

二阶常系数齐次线性方程#

形式：$y'' + py' + qy = 0$ 特征方程：$r^2 + pr + q = 0$ 通解：

1. 两个不等实根 $r_1, r_2$：$y = C_1e^{r_1x} + C_2e^{r_2x}$
2. 两个相等实根 $r_1 = r_2$：$y = (C_1 + C_2x)e^{r_1x}$
3. 共轭复根 $r = \alpha \pm i\beta$：$y = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$

指数函数#

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + o(x^n)$

三角函数#

正弦函数： $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$

余弦函数： $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (-1)^n \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$

对数函数#

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

二项式展开#

$(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots + \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}x^n + o(x^n)$

几何级数#

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

其他重要展开#

反正切函数： $\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots + (-1)^n \frac{x^{2n+1}}{2n+1} + o(x^{2n+2})$

反正弦函数： $\arcsin x = x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + \cdots + o(x^{2n+1})$

微分公式与法则#

• $dy = f'(x)dx$
• $d(C) = 0$
• $d(x^\mu) = \mu x^{\mu-1} dx$
• $d(\sin x) = \cos x dx$
• $d(\cos x) = -\sin x dx$
• $d(e^x) = e^x dx$
• $d(\ln x) = \frac{1}{x} dx$

高阶导数公式#

• $(x^n)^{(n)} = n!$
• $(\sin x)^{(n)} = \sin\left(x + \frac{n\pi}{2}\right)$
• $(\cos x)^{(n)} = \cos\left(x + \frac{n\pi}{2}\right)$
• $(e^x)^{(n)} = e^x$
• $(\ln x)^{(n)} = (-1)^{n-1} \frac{(n-1)!}{x^n}$
• 莱布尼茨公式：$(uv)^{(n)} = \sum\limits_{k=0}^{n} C_n^k u^{(n-k)} v^{(k)}$

微分中值定理#

• 罗尔定理：若 $f(x)$ 在 $[a,b]$ 连续，$(a,b)$ 可导，且 $f(a) = f(b)$，则存在 $\xi \in (a,b)$，使 $f'(\xi) = 0$
• 拉格朗日中值定理：若 $f(x)$ 在 $[a,b]$ 连续，$(a,b)$ 可导，则存在 $\xi \in (a,b)$，使 $f(b) - f(a) = f'(\xi)(b-a)$
• 柯西中值定理：若 $f(x), g(x)$ 在 $[a,b]$ 连续，$(a,b)$ 可导，且 $g'(x) \neq 0$，则存在 $\xi \in (a,b)$，使 $\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)}$

函数单调性与凹凸性#

• 单调性判别：若 $f'(x) > 0$，则 $f(x)$ 单调递增；若 $f'(x) < 0$，则 $f(x)$ 单调递减
• 凹凸性判别：若 $f''(x) > 0$，则 $f(x)$ 凹；若 $f''(x) < 0$，则 $f(x)$ 凸