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

\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

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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

\int \frac{dx}{\sqrt{x^2 + a^2}} = \operatorname{arsinh} \frac{x}{a} + C

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

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

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

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

推导方法

\begin{align}
\int \frac{dx}{\sqrt{x^2 + a^2}}
    &= \int \frac{a \cosh t}{a \cosh t} \, dt && (x = a \sinh t) \\
    &= \int dt \\
    &= t + C \\
    &= \ln\left|x + \sqrt{x^2 + a^2}\right| + C
\end{align}

\begin{align}
\int \frac{dx}{\sqrt{x^2 - a^2}}
    &= \int \frac{a \sinh t}{a \sinh t} \, dt && (x = a \cosh t) \\
    &= \int dt \\
    &= t + C \\
    &= \ln\left|x + \sqrt{x^2 - a^2}\right| + C
\end{align}

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

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

\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

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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

\begin{align}
\int \sqrt{x^2 + a^2} \, dx
    &= \int a \cosh t \cdot a \cosh t \, dt = a^2 \int \cosh^2 t \, dt && (x = a \sinh t) \\
    &= a^2 \int \frac{\cosh 2t + 1}{2} \, dt = \frac{a^2}{2}\left(\frac{\sinh 2t}{2} + t\right) + C \\
    &= \frac{a^2}{2}(\sinh t \cosh t + 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{align}

\begin{align}
\int \sqrt{x^2 - a^2} \, dx
    &= \int a \sinh t \cdot a \sinh t \, dt = a^2 \int \sinh^2 t \, dt && (x = a \cosh t) \\
    &= a^2 \int \frac{\cosh 2t - 1}{2} \, dt = \frac{a^2}{2}\left(\frac{\sinh 2t}{2} - t\right) + C \\
    &= \frac{a^2}{2}(\sinh t \cosh t - 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{align}

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

推广形式

\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

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要点 05 - 三角函数积分

降幂公式

\sin^2 x = \frac{1 - \cos 2x}{2}

\cos^2 x = \frac{1 + \cos 2x}{2}

\sin^3 x = \frac{3\sin x - \sin 3x}{4}

\cos^3 x = \frac{3\cos x + \cos 3x}{4}

基本积分

\int \sin x \, dx = -\cos x + C

\int \cos x \, dx = \sin x + C

\int \tan x \, dx = -\ln|\cos x| + C

\int \cot x \, dx = \ln|\sin x| + C

万能代换

令 $t = \tan\frac{x}{2}$，则：

\sin x = \frac{2t}{1 + t^2}, \quad \cos x = \frac{1 - t^2}{1 + t^2}, \quad dx = \frac{2 \, dt}{1 + t^2}

适用类型：$R(\sin x, \cos x)$（有理函数形式）

常用结论

\int \sin^n x \, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x \, dx

\int \cos^n x \, dx = \frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2} x \, dx

积化和差

\sin A \cos B = \frac{1}{2}\sin(A+B) + \frac{1}{2}\sin(A-B)

\cos A \cos B = \frac{1}{2}\cos(A+B) + \frac{1}{2}\cos(A-B)

\sin A \sin B = \frac{1}{2}\cos(A-B) - \frac{1}{2}\cos(A+B)

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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
• 三角函数积分（降幂、万能代换、积化和差）