微积分上册常用公式

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微积分上册常用公式

2025-01-02

大学数学

微积分

/

数学

1. 常见等价无穷小#

\begin{aligned} \large ln\left(1+x\right)&\large\sim\ x \\\\ \large log_a\left(1+x\right)&\large\sim\frac{x}{lna} \\\\ \large1-cosx &\large\sim\frac{1}{2}x^2 \\\\ \large x-sin{x}&\large\sim\frac{1}{6}x^3 \\\\ \large\left(1+x\right)^\alpha-1&\large\sim\alpha\ x \\\\ \large x-arctan{x}&\large\sim\frac{1}{3}x^3 \\\\ \large e^x-1&\large\sim\ x \\\\ \large a^x-1&\large\sim\ xlna \\\\ \large tan{x}-x&\large\sim\frac{1}{6}x^3 \\\\ \large arcsin{x}-x&\large\sim\frac{1}{6}x^3 \\\\ \large tan{x}-sin{x}&\large \sim\frac{1}{2}x^3 \end{aligned}

2. 高中大概率没讲过的导数#

\begin{aligned} \large \left(arctan{x}\right)^\prime&\large=\frac{1}{1+x^2} \\\\ \large \left(arcsin{x}\right)^\prime&\large=\frac{1}{\sqrt{1-x^2}} \\\\ \large \left(arccos{x}\right)^\prime&\large=\frac{-1}{\sqrt{1-x^2}} \\\\ \large \left(cot{x}\right)^\prime&\large=\frac{-1}{sin^2x} \end{aligned}

3. 高阶导数#

\begin{aligned} \large\left(x^\alpha\right)^{\left(n\right)}&\large=\alpha\left(\alpha-1\right)\ldots\left(\alpha-n+1\right)x^{\alpha-n} \\\\ \large \left(\frac{1}{x}\right)^{\left(n\right)}&\large=\frac{\left(-1\right)^nn!}{x^{n+1}} \\\\ \large \left(a^x\right)^{\left(n\right)}&\large=\left(lna\right)^na^x \\\\ \large \left(sin{x}\right)^{\left(n\right)}&\large=sin{\left(x+\frac{n\pi}{2}\right)} \\\\ \large \left(cos{x}\right)^{\left(n\right)}&\large=cos{\left(x+\frac{n\pi}{2}\right)} \end{aligned}

高阶导莱布尼茨公式#

用于解决一些两式相乘求高阶导，k大于等于某值后其中一导数为零

$\large\left(f\left(x\right)g\left(x\right)\right)^{\left(n\right)}=\sum\limits_{k=0}^{n}C_n^kf\left(x\right)^{\left(k\right)}g\left(x\right)^{\left(n-k\right)}$

4. 曲率#

\begin{aligned} \large K=\frac{ \left|y^{\prime\prime}\right|}{\left(1+y^{\prime2}\right)^\frac{3}{2}} \end{aligned}

曲率半径 $\large R=\frac{\large 1}{\large K}$

5. 不定积分#

\begin{aligned} \large \int\frac{1}{sin^2x}dx&\large=\int{csc}^2{x}dx=-cot{x}+C \\\\ \large \int{\frac{1}{\sqrt{x^2+a^2}}dx}&\large=ln\left(x+\sqrt{x^2+a^2}\right)+C \\\\ \large \int{sec{x}dx}&\large=ln\left|sec{x}+tan{x}\right|+C \\\\ \large \int{\frac{1}{\sqrt{x^2-a^2}}dx}&\large=ln\left|x+\sqrt{x^2-a^2}\right|+C \\\\ \large \int s e c{x}tan{x}dx&\large=sec{x}+C \\\\ \large \int c o t{x}dx&\large=ln\left|sin{x}\right|+C \\\\ \large \int c s c{x}dx&\large=ln\left|csc{x}-cot{x}\right|+C \\\\ \large \int c s c{x}cot{x}dx&\large=-csc{x}+C \end{aligned}

6. 积分求导法则#

$\large \left(\int_{a}^{g\left(x\right)}f\left(t\right)dt\right)^{\prime}=f\left(g\left(x\right)\right)g^\prime\left(x\right)\ \ \ \text{a为常数}$

微积分上册常用公式

https://a1kari8.github.io/posts/calculus1_common_formulas/

作者

A1kari8

发布于

2025-01-02

许可协议

CC BY-NC-SA 4.0

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