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}{x^2 + a^2} 型积分

基本公式

\int \frac{dx}{x^2 + a^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}{x^2 + a^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}{x^2 + a^2} &= \frac{1}{2} \ln(x^2 + a^2) + C \\
\int \frac{dx}{(x^2 + a^2)^2} &= \frac{x}{2a^2(x^2 + a^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 - \sqrt{x^2 \pm a^2} 型积分

基本公式

\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 \quad (|x| > |a|)

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

推导方法

\sqrt{x^2 + a^2} 型：令 $x = a \tan t$，则 $dx = a \sec^2 t , dt$，\sqrt{x^2 + a^2} = a \sec t

\begin{aligned}
\int \sqrt{x^2 + a^2} \, dx &= \int a \sec t \cdot a \sec^2 t \, dt = a^2 \int \sec^3 t \, dt \\
&= a^2 \cdot \frac{1}{2}(\sec t \tan t + \ln|\sec t + \tan t|) + C \\
&= \frac{a^2}{2}\left(\frac{x}{a}\sqrt{1 + \frac{x^2}{a^2}} + \ln\left|\frac{x}{a} + \sqrt{1 + \frac{x^2}{a^2}}\right|\right) + C \\
&= \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\ln\left|x + \sqrt{x^2 + a^2}\right| + C
\end{aligned}

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

\begin{aligned}
\int \sqrt{x^2 - a^2} \, dx &= \int a \tan t \cdot a \sec t \tan t \, dt = a^2 \int \sec t \tan^2 t \, dt \\
&= a^2 \int \sec t (\sec^2 t - 1) \, dt = a^2 \int (\sec^3 t - \sec t) \, dt \\
&= a^2 \left[\frac{1}{2}(\sec t \tan t + \ln|\sec t + \tan t|) - \ln|\sec t + \tan t|\right] + C \\
&= \frac{a^2}{2}\sec t \tan t - \frac{a^2}{2}\ln|\sec t + \tan t| + C \\
&= \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln\left|x + \sqrt{x^2 - a^2}\right| + C
\end{aligned}

\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

\begin{aligned}
\int \sqrt{a^2 - x^2} \, dx &= \int a \cos t \cdot a \cos t \, dt = a^2 \int \cos^2 t \, dt \\
&= a^2 \int \frac{1 + \cos 2t}{2} \, dt = \frac{a^2}{2}\left(t + \frac{\sin 2t}{2}\right) + C \\
&= \frac{a^2}{2}\left(\arcsin\frac{x}{a} + \frac{x}{a}\sqrt{1 - \frac{x^2}{a^2}}\right) + C \\
&= \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C
\end{aligned}

推广形式

\begin{aligned}
\int (x + b)\sqrt{x^2 + a^2} \, dx &= \frac{1}{3}(x + b)(x^2 + a^2)^{3/2} - \frac{b}{2}x\sqrt{x^2 + a^2} - \frac{ab^2}{2}\ln\left|x + \sqrt{x^2 + a^2}\right| + C \\
\int x\sqrt{x^2 + a^2} \, dx &= \frac{1}{3}(x^2 + a^2)^{3/2} + C \\
\int x\sqrt{x^2 - a^2} \, dx &= \frac{1}{3}(x^2 - a^2)^{3/2} + C \\
\int x\sqrt{a^2 - x^2} \, dx &= -\frac{1}{3}(a^2 - x^2)^{3/2} + 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}{x^2 + a^2} 型积分公式
• \frac{1}{\sqrt{x^2 \pm a^2}} 型积分公式
• \sqrt{x^2 \pm a^2} 型积分公式