postgraduate-prep/subjects/math/04_积分.md

笔记记录

要点 01 - 积分与极限求和式的转化

根据公式

\int_a^b f(x)dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x

对于均匀矩形分割的情况，实际上只用分离出 \frac{1}{n}

\int_a^b f(x)dx = \lim_{n\to\infty}\sum_{i=1}^n f\left(a + \frac{(b-a) i}{n}\right) \frac{b-a}{n}

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要点 02 - \frac{1}{a^2 + x^2} 型积分

基本公式

\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan \frac{x}{a} + C \quad (a > 0)

推导（换元法）：令 $x = a \tan t$，则 dx = a \sec^2 t \, dt

\int \frac{dx}{a^2 + x^2} = \int \frac{a \sec^2 t}{a^2 + a^2 \tan^2 t} \, dt = \int \frac{a \sec^2 t}{a^2 \sec^2 t} \, dt = \int \frac{dt}{a} = \frac{t}{a} + C = \frac{1}{a} \arctan \frac{x}{a} + C

推广形式

\begin{aligned}
\int \frac{dx}{x^2 + a^2} &= \frac{1}{a} \arctan \frac{x}{a} + C \\
\int \frac{dx}{b^2 + (x + c)^2} &= \frac{1}{b} \arctan \frac{x + c}{b} + C \\
\int \frac{x \, dx}{a^2 + x^2} &= \frac{1}{2} \ln(a^2 + x^2) + C \\
\int \frac{dx}{(a^2 + x^2)^2} &= \frac{x}{2a^2(a^2 + x^2)} + \frac{1}{2a^3} \arctan \frac{x}{a} + C
\end{aligned}$$

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### 要点 03 - $\frac{1}{\sqrt{x^2 \pm a^2}}$ 型积分

#### 基本公式

$$\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left|x + \sqrt{x^2 + a^2}\right| + C = \operatorname{arsinh} \frac{x}{a} + C$$

$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left|x + \sqrt{x^2 - a^2}\right| + C = \operatorname{arcosh} \frac{x}{a} + C \quad (|x| > |a|)$$

$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C = -\arccos \frac{x}{a} + C \quad (|x| < |a|)$$

#### 推导方法

**$\sqrt{x^2 + a^2}$ 型**：令 $x = a \sinh t$，则 $dx = a \cosh t \, dt$，$\sqrt{x^2 + a^2} = a \cosh t$

$$\int \frac{dx}{\sqrt{x^2 + a^2}} = \int \frac{a \cosh t}{a \cosh t} \, dt = \int dt = t + C = \ln\left|x + \sqrt{x^2 + a^2}\right| + C$$

**$\sqrt{x^2 - a^2}$ 型**：令 $x = a \cosh t$（$x > a$），则 $dx = a \sinh t \, dt$，$\sqrt{x^2 - a^2} = a \sinh t$

$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \int \frac{a \sinh t}{a \sinh t} \, dt = \int dt = t + C = \ln\left|x + \sqrt{x^2 - a^2}\right| + C$$

**$\sqrt{a^2 - x^2}$ 型**：令 $x = a \sin t$，则 $dx = a \cos t \, dt$，$\sqrt{a^2 - x^2} = a \cos t$

$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \int \frac{a \cos t}{a \cos t} \, dt = \int dt = t + C = \arcsin \frac{x}{a} + C$$

#### 等效形式

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

#### 推广形式

$$\begin{aligned}
\int \frac{dx}{\sqrt{(x + b)^2 + a^2}} &= \ln\left|x + b + \sqrt{(x + b)^2 + a^2}\right| + C \\
\int \frac{dx}{\sqrt{(x + b)^2 - a^2}} &= \ln\left|x + b + \sqrt{(x + b)^2 - a^2}\right| + C \quad (|x + b| > |a|) \\
\int \frac{x \, dx}{\sqrt{x^2 + a^2}} &= \sqrt{x^2 + a^2} + C \\
\int \frac{x \, dx}{\sqrt{x^2 - a^2}} &= \sqrt{x^2 - a^2} + C \\
\int \sqrt{x^2 + a^2} \, dx &= \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\ln\left|x + \sqrt{x^2 + a^2}\right| + C \\
\int \sqrt{x^2 - a^2} \, dx &= \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln\left|x + \sqrt{x^2 - a^2}\right| + C
\end{aligned}$$

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### 要点 04 - sec x、csc x 的积分

#### 基本积分公式

$$\int \sec x \, dx = \ln|\sec x + \tan x| + C$$

**推导方法**（分子分母策略）：

$$\int \sec x \, dx = \int \sec x \cdot \frac{\sec x + \tan x}{\sec x + \tan x} \, dx = \int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, dx = \ln|\sec x + \tan x| + C$$

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$$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$$

**推导方法**（类似地）：

$$\int \csc x \, dx = \int \csc x \cdot \frac{\csc x - \cot x}{\csc x - \cot x} \, dx = \int \frac{\csc^2 x - \csc x \cot x}{\csc x - \cot x} \, dx = -\ln|\csc x + \cot x| + C$$

#### 其他常用积分

$$\begin{aligned}
\int \sec^2 x \, dx &= \tan x + C \\
\int \csc^2 x \, dx &= -\cot x + C \\
\int \sec x \tan x \, dx &= \sec x + C \\
\int \csc x \cot x \, dx &= -\csc x + C \\
\int \sec^3 x \, dx &= \frac{1}{2}(\sec x \tan x + \ln|\sec x + \tan x|) + C \\
\int \csc^3 x \, dx &= \frac{1}{2}(-\csc x \cot x + \ln|\csc x + \cot x|) + C
\end{aligned}$$

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### 知识点
- 定积分的定义
- 黎曼和与积分的关系
- 均匀分割技巧
- $\frac{1}{a^2 + x^2}$ 型积分公式
- $\frac{1}{\sqrt{x^2 \pm a^2}}$ 型积分公式
- sec x、csc x 的积分公式