Update 公式大全.md
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@ -45,6 +45,16 @@ date: 2022-12-24T13:05:40+08:00
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- [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)
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- [不定积分的基本性质](#不定积分的基本性质)
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- [不定积分的基本积分公式](#不定积分的基本积分公式)
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- [不定积分计算方法](#不定积分计算方法)
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- [第一类换元法(凑微分法)](#第一类换元法凑微分法)
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- [第二类换元法(变量置换法)](#第二类换元法变量置换法)
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- [三角代换](#三角代换)
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- [根式代换](#根式代换)
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- [倒代换](#倒代换)
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- [指数代换](#指数代换)
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- [万能代换](#万能代换)
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- [组合积分法](#组合积分法)
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- [值得记忆的定积分公式](#值得记忆的定积分公式)
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- [线性代数](#线性代数)
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- [行列式](#行列式)
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- [行列式定义和性质](#行列式定义和性质)
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@ -310,6 +320,116 @@ date: 2022-12-24T13:05:40+08:00
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13. $\int {\frac{1}{\sqrt{a^2-x^2}}}dx=\arcsin \frac{x}{a} +C(a>0)$,$\int {\frac{1}{\sqrt{1-x^2}}}dx = \arctan x +C$
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14. $\int {\frac{dx}{a^2-x^2}}dx=\frac{1}{2a}\ln |\frac{a+x}{a-x}| +C$
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15. $\int {\frac{1}{\sqrt{x^2 \pm a^2}}}dx=\ln |x+\sqrt{x^2 \pm a^2}| +C$
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16. $\int e^{ax}\cos bx dx = \frac{e^{ax}}{a^2+b^2}\left[a\cos bx + b \sin bx \right] + C$
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17. $\int e^{ax} \sin bx dx = \frac{e^{ax}}{a^2+b^2}\left[a\sin bx + b \cos bx \right] + C$
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#### 不定积分计算方法
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#### 第一类换元法(凑微分法)
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常见凑微分
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1. $\int (x^n)\cdot x^{n-1} dx = \frac{1}{n} \int f(x^n)dx^n$
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2. $\int f(\sqrt{x^3} ) \cdot \sqrt{x}dx = \frac{2}{3}\int f(\sqrt{x^3})d\sqrt{x^3}$
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3. $\int(x+\sin x) \cdot (1+\cos x) dx = \int f(x+ \sin x)d(x+\sin x)$
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4. $\int \frac{g(x)}{(1+x)^2}dx = \int g(x)d(-\frac{1}{1+x})$
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5. $\int \frac{g(x)}{1-x^2}dx = \int g(x)d(\ln \frac{1+x}{1-x})$
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6. $\int f(ax^2+bx+c) \cdot (2ax+b)dx = \int f(ax^2+bx+c)d(ax^2+bx+c)$
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7. $\int g(x) \cdot e^x(1+x)dx = \int g(x)d(xe^x)$
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8. $\int f(x+\frac{1}{x}) \cdot (1-\frac{1}{x^2})dx = \int xf(x+\frac{1}{x})d(x+\frac{1}{x})$
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9. $\int f(x+\frac{1}{x}) \cdot (x-\frac{1}{x})dx = \int xf(x+\frac{1}{x})d(x+\frac{1}{x})$
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10. $\int g(x) \cdot \frac{1}{\sqrt{1+x^2}}dx = \int g(x)d\ln(x+\sqrt{1+x^2})$
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11. $\int g(x) \cdot \frac{1}{1+x^2}dx = \int g(x)d\arctan x = -\int g(x)d\arctan \frac{1}{x}$
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12. $\int f(1-\frac{1}{x}) \cdot \frac{1}{x(x-1)}dx = \int f(1-\frac{1}{x}) \cdot \frac{1}{1-\frac{1}{x}} \cdot \frac{1}{x^2}dx = \int f(1-\frac{1}{x})d\ln(1-\frac{1}{x})$
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##### 第二类换元法(变量置换法)
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**_计算后必须回代_**
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###### 三角代换
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1. 若$R(-\sin x , \cos x) = -R(\sin x,\cos x)$,令$u=\cos x$或凑微分$d\cos x$
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2. 若$R(\sin x , -\cos x) = -R(\sin x,\cos x)$,令$u=\sin x$或凑微分$d\sin x$
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3. 若$R(-\sin x , -\cos x) = R(\sin x,\cos x)$,令$u=\tan x$或凑微分$d\tan x$
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或
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1. 被积函数含有$\sqrt{a^2-x^2}$,令$x=a\sin t (或a \cos t)$
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2. 被积函数含有$\sqrt{a^2+x^2}$,令$x=a\tan t $
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3. 被积函数含有$\sqrt{x^2 - a^2}$,令$x=a\sin t$
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###### 根式代换
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对根式进行代换
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如$\int\frac{\sqrt{x}}{1+\sqrt{x}^3}$
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取$t=x^{\frac{1}{6}},t^6 = x => dx=6t^5dt$
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###### 倒代换
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将x替换为对应倒数(通常分母次数比分子高)
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例如$\int \frac{1}{x^2 \sqrt{x^2+1}}dx$
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令$x = \frac{1}{t}$,则$dx = -\frac{dt}{t^2}$
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###### 指数代换
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将指数替换为t
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例如$\int \frac{1}{\sqrt{1+e^x}}dx$
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$\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}} $
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###### 万能代换
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对于形如$\int R(\cos x ,\sin x)dx$的不定积分
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可换元
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$$
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\cos x = \frac{1-t^2}{1+t^2}
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$$
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$$
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\sin x = \frac{2t}{1+t^2}
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$$
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原积分转化为$\int R(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})\frac{2}{1+t^2}dt$
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##### 组合积分法
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对于
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$$
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\int \frac{dx}{(ax+b)(mx+n)}
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$$
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可令
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$$
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I=\int \frac{dx}{(ax+b)(mx+n)}
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$$
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$$
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J=\int \frac{xdx}{(ax+b)(mx+n)}
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$$
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$$
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\begin{cases}
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bI+aJ = \int \frac{(ax+b)dx}{(ax+b)(mx+n)} = \frac{1}{m} \ln|mx+n|+C_1 \\\\
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nI+mJ = \int \frac{(mx+n)dx}{(ax+b)(mx+n)} = \frac{1}{a} \ln|ax+b|+C_2
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\end{cases}
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$$
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$$
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I = \frac{1}{bm-an} \ln|\frac{mx+n}{ax+b}| +C
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$$
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#### 值得记忆的定积分公式
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---
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