积分学

不定积分记得带上常数项！

极坐标相关公式

弧长：

s=∫αβ[r(θ)]2+[r′(θ)]2dθs=\int_{\alpha}^{\beta}\sqrt{ \big[r(\theta)\big]^2+\big[r'(\theta)\big]^2 } \text d\theta

面积：

S=12∫αβ∣r12(θ)−r22(θ)∣dθS=\dfrac{1}{2}\int_{\alpha}^{\beta} {\Big| r_1^2(\theta)- r_2^2(\theta) \Big|} \text d\theta

侧面积（绕x轴）：

S=∫αβ2π∣r(θ)sin⁡θ∣[r(θ)]2+[r′(θ)]2dθS=\int_{\alpha}^{\beta}2\pi \Big| r(\theta)\sin\theta \Big| \sqrt{\left[r(\theta)\right]^{2}+\left[r^{\prime}(\theta)\right]^{2}}\mathrm{d}\theta

旋转体体积（绕极轴/x轴）：

V=2π3∫αβr3(θ)sin⁡θdθV=\frac{2\pi}{3}\int_{\alpha}^{\beta}r^{3}(\theta)\sin\theta\mathrm{d}\theta

导数：

r=r(θ)→x=rcos⁡θ, y=rsin⁡θdydx=dy/dθdx/dθ\begin{aligned} r& =r(\theta) \to \\ x&=r\cos \theta ,\ y=r\sin \theta \\ \dfrac{\text{d}y}{\text{d}x} &= \dfrac{\text{d}y/\text{d}\theta}{\text{d}x/\text{d}\theta} \end{aligned}

参数方程相关

侧面积（绕x轴）：

S=∫αβ2π∣y(t)∣[x′(t)]2+[y′(t)]2dtS=\int_{\alpha}^{\beta}2\pi \Big|y(t) \Big|\sqrt{[x^{\prime}(t)]^{2}+[y^{\prime}(t)]^{2}}\mathrm{d}t

侧面积（绕y轴）：

S=∫αβ2π∣x(t)∣[x′(t)]2+[y′(t)]2dtS=\int_{\alpha}^{\beta}2\pi \Big|x(t) \Big|\sqrt{[x^{\prime}(t)]^{2}+[y^{\prime}(t)]^{2}}\mathrm{d}t

反三角函数相关积分

∫dxa2−x2=arcsin⁡xa+C(a>0)∫dxa2+x2=1aarctan⁡xa+C(a>0)\begin{aligned} \int\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}&=\arcsin\frac{x}{a}+C&(a>0)\\ \int\frac{\mathrm{d}x}{a^2+x^2}&=\textcolor{red}{\frac{1}{a}}\arctan\frac{x}{a}+C&(a>0) \end{aligned}

奇怪的三角函数积分

∫sec⁡xdx=ln⁡∣sec⁡x+tan⁡x∣+C∫sec⁡3xdx=12sec⁡x⋅tan⁡x+12ln⁡∣sec⁡x+tan⁡x∣+C∫csc⁡xdx=ln⁡∣csc⁡x−cot⁡x∣+C∫csc⁡3xdx=−12csc⁡x⋅cot⁡x+12ln⁡∣csc⁡x−cot⁡x∣+C \begin{aligned} \int\sec x\mathrm{d}x &=\ln\left|\sec x+\tan x\right|+C\\ \int\sec^3x\mathrm{d}x&=\frac{1}{2}\sec x\cdot\tan x+\frac{1}{2}\ln\left|\sec x+\tan x\right|+C\\\\ \int\csc x\mathrm{d}x &=\ln\left|\csc x\textcolor{red}{-}\cot x\right|+C\\ \int\csc^3x\mathrm{d}x&=\textcolor{red}{-}\frac12\csc x\cdot\cot x+\frac12\ln\left|\csc x-\cot x\right|+C\ \end{aligned}

华里士公式

∫0π2sin⁡nxdx=∫0π2cos⁡nxdx={n−1n⋅n−3n−2⋯⋯12⋅π2,n=2,4,6,⋯n−1n⋅n−3n−2⋯⋯23⋅1,n=3,5,7,⋯\begin{aligned} \int_0^{\frac{\pi}{2}}\sin^nx\mathrm{d}x=\int_0^{\frac{\pi}{2}}\cos^nx\mathrm{d}x=\begin{cases}\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\cdots\frac{1}{2}\cdot\frac{\pi}{2},&n=2,4,6,\cdots\\\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\cdots\frac{2}{3}\cdot1,&n=3,5,7,\cdots\end{cases} \end{aligned}

n=0时就是pi/2，n=1时就是正常积分。带进去看一下就知道用哪一个了。

我们还能继续拓展到π上：

∫0πsin⁡nxdx={2⋅n−1n⋅n−3n−2⋯⋯12⋅π2,n=2,4,6,⋯2⋅n−1n⋅n−3n−2⋯⋯23⋅1,n=3,5,7,⋯∫0πcos⁡nxdx={2⋅n−1n⋅n−3n−2⋯⋯12⋅π2,n=2,4,6,⋯0,n=1,3,5,7,⋯\begin{aligned} \int_0^{\pi}\sin^nx\mathrm{d}x&=\begin{cases}2\cdot\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\cdots\frac{1}{2}\cdot\frac{\pi}{2},&n=2,4,6,\cdots\\2\cdot \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\cdots\frac{2}{3}\cdot1,&n=3,5,7,\cdots\end{cases}\\\\ \int_0^{\pi}\cos^nx\mathrm{d}x&=\begin{cases}2\cdot\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\cdots\frac{1}{2}\cdot\frac{\pi}{2},&n=2,4,6,\cdots\\0,&n=\textcolor{red}{1},3,5,7,\cdots\end{cases} \end{aligned}

形心公式

x‾=∫abdx∫0f(x)xdy∫abdx∫0f(x)dy=∫abxf(x)dx∫abf(x)dxy‾=∫abdx∫0f(x)ydy∫abdx∫0f(x)dy=12∫abf2(x)dx∫abf(x)dx.\begin{aligned} \overline{x}&=\frac{\int_{a}^{b}\mathrm{d}x\int_{0}^{f(x)}x\mathrm{d}y}{\int_{a}^{b}\mathrm{d}x\int_{0}^{f(x)}\mathrm{d}y}=\frac{\int_{a}^{b}xf(x)\mathrm{d}x}{\int_{a}^{b}f(x)\mathrm{d}x}\\ \overline{y}&=\frac{\int_{a}^{b}\mathrm{d}x\int_{0}^{f(x)}y\mathrm{d}y}{\int_{a}^{b}\mathrm{d}x\int_{0}^{f(x)}\mathrm{d}y}=\frac{\dfrac{1}{2}\int_{a}^{b}f^{2}(x)\mathrm{d}x}{\int_{a}^{b}f(x)\mathrm{d}x}. \end{aligned}

微分学

反函数的导数

dxdy=1y′d2xdy2=−y′′(y′)3\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}y}&=\frac1{y^{\prime}}\\ \frac{\mathrm{d}^2x}{\mathrm{d}y^2}&=\textcolor{red}{-}\frac{y^{\prime\prime}}{(y^{\prime})^3} \end{aligned}

莱布尼茨公式

(uv)(n)=∑k=0nCnku(k)v(n−k)(uv) ^{(n)}= \sum _{k=0}^n C_n^k u^{(k)} v^{(n-k)}

矩阵

分块矩阵的行列式

∣OBmxmCnxnO∣=(−1)m⋅n∣B∣⋅∣C∣\left | \begin{array}{llll} \mathbf{O}&\mathbf{B}_{\text{mxm}}\\ \mathbf{C}_{\text{nxn}}&\mathbf{O} \end{array}\right | =(-1)^{m\sdot n} \big| \mathbf{B} \big| \sdot \big| \mathbf{C} \big|