From d2c1a18e9833498663c8a20cb4eddca45953e3bf Mon Sep 17 00:00:00 2001 From: InkSoul Date: Wed, 20 Sep 2023 21:43:33 +0800 Subject: [PATCH] =?UTF-8?q?Update=20=E5=85=AC=E5=BC=8F=E5=A4=A7=E5=85=A8.m?= =?UTF-8?q?d?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/mathematics/公式大全.md | 120 ++++++++++++++++++++++++++++++++ 1 file changed, 120 insertions(+) diff --git a/content/mathematics/公式大全.md b/content/mathematics/公式大全.md index d96a9c1..07121a7 100644 --- a/content/mathematics/公式大全.md +++ b/content/mathematics/公式大全.md @@ -45,6 +45,16 @@ date: 2022-12-24T13:05:40+08:00 - [原函数和不定积分的基本概念](#原函数和不定积分的基本概念) - [不定积分的基本性质](#不定积分的基本性质) - [不定积分的基本积分公式](#不定积分的基本积分公式) + - [不定积分计算方法](#不定积分计算方法) + - [第一类换元法（凑微分法）](#第一类换元法凑微分法) + - [第二类换元法（变量置换法）](#第二类换元法变量置换法) + - [三角代换](#三角代换) + - [根式代换](#根式代换) + - [倒代换](#倒代换) + - [指数代换](#指数代换) + - [万能代换](#万能代换) + - [组合积分法](#组合积分法) + - [值得记忆的定积分公式](#值得记忆的定积分公式) - [线性代数](#线性代数) - [行列式](#行列式) - [行列式定义和性质](#行列式定义和性质) @@ -310,6 +320,116 @@ date: 2022-12-24T13:05:40+08:00 13. $\int {\frac{1}{\sqrt{a^2-x^2}}}dx=\arcsin \frac{x}{a} +C(a>0)$，$\int {\frac{1}{\sqrt{1-x^2}}}dx = \arctan x +C$ 14. $\int {\frac{dx}{a^2-x^2}}dx=\frac{1}{2a}\ln |\frac{a+x}{a-x}| +C$ 15. $\int {\frac{1}{\sqrt{x^2 \pm a^2}}}dx=\ln |x+\sqrt{x^2 \pm a^2}| +C$ +16. $\int e^{ax}\cos bx dx = \frac{e^{ax}}{a^2+b^2}\left[a\cos bx + b \sin bx \right] + C$ +17. $\int e^{ax} \sin bx dx = \frac{e^{ax}}{a^2+b^2}\left[a\sin bx + b \cos bx \right] + C$ + + +#### 不定积分计算方法 + +#### 第一类换元法（凑微分法） + +常见凑微分 + +1. $\int (x^n)\cdot x^{n-1} dx = \frac{1}{n} \int f(x^n)dx^n$ +2. $\int f(\sqrt{x^3} ) \cdot \sqrt{x}dx = \frac{2}{3}\int f(\sqrt{x^3})d\sqrt{x^3}$ +3. $\int(x+\sin x) \cdot (1+\cos x) dx = \int f(x+ \sin x)d(x+\sin x)$ +4. $\int \frac{g(x)}{(1+x)^2}dx = \int g(x)d(-\frac{1}{1+x})$ +5. $\int \frac{g(x)}{1-x^2}dx = \int g(x)d(\ln \frac{1+x}{1-x})$ +6. $\int f(ax^2+bx+c) \cdot (2ax+b)dx = \int f(ax^2+bx+c)d(ax^2+bx+c)$ +7. $\int g(x) \cdot e^x(1+x)dx = \int g(x)d(xe^x)$ +8. $\int f(x+\frac{1}{x}) \cdot (1-\frac{1}{x^2})dx = \int xf(x+\frac{1}{x})d(x+\frac{1}{x})$ +9. $\int f(x+\frac{1}{x}) \cdot (x-\frac{1}{x})dx = \int xf(x+\frac{1}{x})d(x+\frac{1}{x})$ +10. $\int g(x) \cdot \frac{1}{\sqrt{1+x^2}}dx = \int g(x)d\ln(x+\sqrt{1+x^2})$ +11. $\int g(x) \cdot \frac{1}{1+x^2}dx = \int g(x)d\arctan x = -\int g(x)d\arctan \frac{1}{x}$ +12. $\int f(1-\frac{1}{x}) \cdot \frac{1}{x(x-1)}dx = \int f(1-\frac{1}{x}) \cdot \frac{1}{1-\frac{1}{x}} \cdot \frac{1}{x^2}dx = \int f(1-\frac{1}{x})d\ln(1-\frac{1}{x})$ + +##### 第二类换元法（变量置换法） + +**_计算后必须回代_** + +###### 三角代换 + +1. 若$R(-\sin x , \cos x) = -R(\sin x,\cos x)$,令$u=\cos x$或凑微分$d\cos x$ +2. 若$R(\sin x , -\cos x) = -R(\sin x,\cos x)$,令$u=\sin x$或凑微分$d\sin x$ +3. 若$R(-\sin x , -\cos x) = R(\sin x,\cos x)$,令$u=\tan x$或凑微分$d\tan x$ + +或 + +1. 被积函数含有$\sqrt{a^2-x^2}$,令$x=a\sin t (或a \cos t)$ +2. 被积函数含有$\sqrt{a^2+x^2}$,令$x=a\tan t $ +3. 被积函数含有$\sqrt{x^2 - a^2}$,令$x=a\sin t$ + +###### 根式代换 + +对根式进行代换 + +如$\int\frac{\sqrt{x}}{1+\sqrt{x}^3}$ + +取$t=x^{\frac{1}{6}},t^6 = x => dx=6t^5dt$ + +###### 倒代换 + +将x替换为对应倒数(通常分母次数比分子高) + +例如$\int \frac{1}{x^2 \sqrt{x^2+1}}dx$ + +令$x = \frac{1}{t}$,则$dx = -\frac{dt}{t^2}$ + +###### 指数代换 + +将指数替换为t + +例如$\int \frac{1}{\sqrt{1+e^x}}dx$ + +$\int \frac{1}{e^x\sqrt{1+e^x}}dx = \int \frac{de^x}{e^x\sqrt{1+e^x}} \underrightarrow{t = e^x} \int \frac{dt}{t\sqrt{1+t}} $ + +###### 万能代换 + +对于形如$\int R(\cos x ,\sin x)dx$的不定积分 + +可换元 + +$$ +\cos x = \frac{1-t^2}{1+t^2} +$$ + +$$ +\sin x = \frac{2t}{1+t^2} +$$ + +原积分转化为$\int R(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})\frac{2}{1+t^2}dt$ + +##### 组合积分法 + +对于 + +$$ +\int \frac{dx}{(ax+b)(mx+n)} +$$ + +可令 + +$$ +I=\int \frac{dx}{(ax+b)(mx+n)} +$$ + +$$ +J=\int \frac{xdx}{(ax+b)(mx+n)} +$$ + +$$ +\begin{cases} + bI+aJ = \int \frac{(ax+b)dx}{(ax+b)(mx+n)} = \frac{1}{m} \ln|mx+n|+C_1 \\\\ + nI+mJ = \int \frac{(mx+n)dx}{(ax+b)(mx+n)} = \frac{1}{a} \ln|ax+b|+C_2 +\end{cases} +$$ + +$$ +I = \frac{1}{bm-an} \ln|\frac{mx+n}{ax+b}| +C +$$ + +#### 值得记忆的定积分公式 + ---