193 lines
4.9 KiB
Markdown
193 lines
4.9 KiB
Markdown
## 笔记记录
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### 要点 02 - 分式型积分（$a, b > 0$）
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#### 基本公式
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$$
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\int \frac{dx}{a^2 x^2 + b^2} = \frac{1}{ab} \arctan \frac{ax}{b} + C
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$$
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**推导**（换元法）：令 $t = \dfrac{a}{b} x$，则 $x = \dfrac{b}{a} t$，$dx = \dfrac{b}{a} dt$
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$$
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\int \frac{dx}{a^2 x^2 + b^2} = \int \frac{\frac{b}{a} dt}{b^2 t^2 + b^2}
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= \frac{1}{ab} \int \frac{dt}{t^2 + 1}
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= \frac{1}{ab} \arctan t + C
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= \frac{1}{ab} \arctan \frac{ax}{b} + C
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$$
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#### 推广形式
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$$
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\int \frac{dx}{a^2 (x + c)^2 + b^2} = \frac{1}{ab} \arctan \frac{a(x + c)}{b} + C
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$$
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$$
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\int \frac{x \, dx}{a^2 x^2 + b^2} = \frac{1}{2a^2} \ln(a^2 x^2 + b^2) + C
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$$
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$$
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\int \frac{dx}{(a^2 x^2 + b^2)^2} = \frac{x}{2b^2(a^2 x^2 + b^2)} + \frac{1}{2ab^3} \arctan \frac{ax}{b} + C
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$$
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---
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### 要点 03 - 根号分式型积分（$a, b > 0$）
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#### 基本公式
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**型 I**：$a^2 x^2 + b^2$
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$$
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\int \frac{dx}{\sqrt{a^2 x^2 + b^2}} = \frac{1}{a} \ln\left|ax + \sqrt{a^2 x^2 + b^2}\right| + C
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= \frac{1}{a} \operatorname{arsinh} \frac{ax}{b} + C
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$$
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**型 II**：$a^2 x^2 - b^2$
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$$
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\int \frac{dx}{\sqrt{a^2 x^2 - b^2}} = \frac{1}{a} \ln\left|ax + \sqrt{a^2 x^2 - b^2}\right| + C \quad (|ax| > |b|)
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= \frac{1}{a} \operatorname{arcosh} \frac{ax}{b} + C
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$$
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**型 III**：$b^2 - a^2 x^2$
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$$
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\int \frac{dx}{\sqrt{b^2 - a^2 x^2}} = \frac{1}{a} \arcsin \frac{ax}{b} + C \quad (|ax| < |b|)
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$$
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#### 推导方法
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令 $t = ax$，则 $x = \dfrac{t}{a}$，$dx = \dfrac{dt}{a}$，化为标准形式后代入已知公式。
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**型 I**（$x = \frac{b}{a} \sinh t$ 或 $t = b \sinh u$）：
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$$
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\begin{align}
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\int \frac{dx}{\sqrt{a^2 x^2 + b^2}}
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&= \frac{1}{a} \int \frac{dt}{\sqrt{t^2 + b^2}}
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= \frac{1}{a} \ln\left|t + \sqrt{t^2 + b^2}\right| + C \\
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&= \frac{1}{a} \ln\left|ax + \sqrt{a^2 x^2 + b^2}\right| + C
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\end{align}
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$$
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**型 II**（$x = \frac{b}{a} \cosh t$）：
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$$
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\begin{align}
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\int \frac{dx}{\sqrt{a^2 x^2 - b^2}}
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&= \frac{1}{a} \int \frac{dt}{\sqrt{t^2 - b^2}}
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= \frac{1}{a} \ln\left|t + \sqrt{t^2 - b^2}\right| + C \quad (|t| > |b|) \\
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&= \frac{1}{a} \ln\left|ax + \sqrt{a^2 x^2 - b^2}\right| + C
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\end{align}
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$$
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**型 III**（$x = \frac{b}{a} \sin t$）：
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$$
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\begin{align}
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\int \frac{dx}{\sqrt{b^2 - a^2 x^2}}
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&= \frac{1}{a} \int \frac{dt}{\sqrt{b^2 - t^2}}
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= \frac{1}{a} \arcsin \frac{t}{b} + C \quad (|t| < |b|) \\
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&= \frac{1}{a} \arcsin \frac{ax}{b} + C
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\end{align}
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$$
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#### 推广形式
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$$
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\int \frac{dx}{\sqrt{a^2 (x + c)^2 + b^2}} = \frac{1}{a} \ln\left|a(x + c) + \sqrt{a^2 (x + c)^2 + b^2}\right| + C
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$$
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$$
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\int \frac{dx}{\sqrt{a^2 (x + c)^2 - b^2}} = \frac{1}{a} \ln\left|a(x + c) + \sqrt{a^2 (x + c)^2 - b^2}\right| + C \quad (|a(x + c)| > |b|)
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$$
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$$
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\int \frac{x \, dx}{\sqrt{a^2 x^2 + b^2}} = \frac{1}{a^2} \sqrt{a^2 x^2 + b^2} + C
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$$
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$$
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\int \frac{x \, dx}{\sqrt{a^2 x^2 - b^2}} = \frac{1}{a^2} \sqrt{a^2 x^2 - b^2} + C
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$$
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---
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### 要点 04 - 根号二次型积分（$a, b > 0$）
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#### 基本公式
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令 $t = ax$，统一化为标准形式后积分。
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**型 I**：$\sqrt{a^2 x^2 + b^2}$
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$$
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\int \sqrt{a^2 x^2 + b^2} \, dx = \frac{x}{2}\sqrt{a^2 x^2 + b^2} + \frac{b^2}{2a} \ln\left|ax + \sqrt{a^2 x^2 + b^2}\right| + C
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$$
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**型 II**：$\sqrt{a^2 x^2 - b^2}$
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$$
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\int \sqrt{a^2 x^2 - b^2} \, dx = \frac{x}{2}\sqrt{a^2 x^2 - b^2} - \frac{b^2}{2a} \ln\left|ax + \sqrt{a^2 x^2 - b^2}\right| + C \quad (|ax| > |b|)
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$$
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**型 III**：$\sqrt{b^2 - a^2 x^2}$
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$$
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\int \sqrt{b^2 - a^2 x^2} \, dx = \frac{x}{2}\sqrt{b^2 - a^2 x^2} + \frac{b^2}{2a} \arcsin\frac{ax}{b} + C \quad (|ax| < |b|)
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$$
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#### 推导方法
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令 $t = ax$，则 $x = \frac{t}{a}$，$dx = \frac{dt}{a}$，化为对 $t$ 的标准形式。
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**型 I**（$t = b \sinh u$）：
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$$
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\begin{align}
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\int \sqrt{a^2 x^2 + b^2} \, dx
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&= \frac{1}{a} \int \sqrt{t^2 + b^2} \, dt
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= \frac{1}{a} \left( \frac{t}{2}\sqrt{t^2 + b^2} + \frac{b^2}{2}\ln\left|t + \sqrt{t^2 + b^2}\right| \right) + C \\
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&= \frac{x}{2}\sqrt{a^2 x^2 + b^2} + \frac{b^2}{2a} \ln\left|ax + \sqrt{a^2 x^2 + b^2}\right| + C
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\end{align}
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$$
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**型 II**（$t = b \cosh u$）：
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$$
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\begin{align}
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\int \sqrt{a^2 x^2 - b^2} \, dx
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&= \frac{1}{a} \int \sqrt{t^2 - b^2} \, dt
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= \frac{1}{a} \left( \frac{t}{2}\sqrt{t^2 - b^2} - \frac{b^2}{2}\ln\left|t + \sqrt{t^2 - b^2}\right| \right) + C \\
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&= \frac{x}{2}\sqrt{a^2 x^2 - b^2} - \frac{b^2}{2a} \ln\left|ax + \sqrt{a^2 x^2 - b^2}\right| + C
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\end{align}
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$$
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**型 III**（$t = b \sin u$）：
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$$
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\begin{align}
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\int \sqrt{b^2 - a^2 x^2} \, dx
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&= \frac{1}{a} \int \sqrt{b^2 - t^2} \, dt
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= \frac{1}{a} \left( \frac{t}{2}\sqrt{b^2 - t^2} + \frac{b^2}{2}\arcsin\frac{t}{b} \right) + C \\
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&= \frac{x}{2}\sqrt{b^2 - a^2 x^2} + \frac{b^2}{2a} \arcsin\frac{ax}{b} + C
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\end{align}
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$$
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#### 推广形式
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$$
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\int x\sqrt{a^2 x^2 + b^2} \, dx = \frac{1}{3a^2}(a^2 x^2 + b^2)^{3/2} + C
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$$
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$$
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\int x\sqrt{a^2 x^2 - b^2} \, dx = \frac{1}{3a^2}(a^2 x^2 - b^2)^{3/2} + C
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$$
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$$
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\int x\sqrt{b^2 - a^2 x^2} \, dx = -\frac{1}{3a^2}(b^2 - a^2 x^2)^{3/2} + C
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$$
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---
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