常用的凑微分公式汇总

一、前言

凑微分的目的就是将积分 $\int \Phi(x) \mathrm{~d} x$ 改写成 $\int f(\phi(x)) \mathrm{~d} \phi(x)$ 的形式，即：

$$
\int \textcolor{orange}{\Phi(x)} \mathrm{~d} x = \int f(\textcolor{lightgreen}{\phi(x)}) \mathrm{~d} \textcolor{lightgreen}{\phi(x)}
$$

经过上述变换，就可以将积分变量从 $x$ 拓展成更复杂的 $\phi(x)$, 从而可以在大多数时候达到简化被积函数的作用。

在本文中，「荒原之梦考研数学」就给同学们汇总了考研数学（高等数学）解题过程中常用的凑微分公式。

二、正文

常用的凑微分公式如下：

$$
\begin{align}
& \int f(\textcolor{lightgreen}{ax + b}) \mathrm{~d} x = \int \frac{1}{a} f(\textcolor{lightgreen}{ax + b}) \mathrm{~d} (\textcolor{lightgreen}{ax + b}) \quad (a \neq 0) \\ \notag \\
& \int f(\textcolor{lightgreen}{x^{\alpha}}) x^{\alpha – 1} \mathrm{~d} x = \frac{1}{\alpha} \int f(\textcolor{lightgreen}{x^{\alpha}}) \mathrm{~d} (\textcolor{lightgreen}{x^{\alpha}}) \quad (\alpha \neq 0) \\ \notag \\
& \int f(\textcolor{lightgreen}{\ln x}) \frac{1}{x} \mathrm{~d} x = \int f(\textcolor{lightgreen}{\ln x}) \mathrm{~d} (\textcolor{lightgreen}{\ln x}) \\ \notag \\
& \int f(\textcolor{lightgreen}{\sin x}) \cos x \mathrm{~d} x = \int f(\textcolor{lightgreen}{\sin x}) \mathrm{~d} (\textcolor{lightgreen}{\sin x}) \\ \notag \\
& \int f(\textcolor{lightgreen}{\cos x}) \sin x \mathrm{~d} x = -\int f(\textcolor{lightgreen}{\cos x}) \mathrm{~d} (\textcolor{lightgreen}{\cos x}) \\ \notag \\
& \int f(\textcolor{lightgreen}{\tan x}) \frac{1}{\cos^{2} x} \mathrm{~d} x = \int f(\textcolor{lightgreen}{\tan x}) \mathrm{~d} (\textcolor{lightgreen}{\tan x}) \\ \notag \\
& \int f(\textcolor{lightgreen}{\arctan x}) \frac{1}{1 + x^{2}} \mathrm{~d} x = \int f(\textcolor{lightgreen}{\arctan x}) \mathrm{~d} (\textcolor{lightgreen}{\arctan x}) \\ \notag \\
& \int f(\textcolor{lightgreen}{\arcsin x}) \frac{1}{\sqrt{1 – x^{2}}} \mathrm{~d} x = \int f(\textcolor{lightgreen}{\arcsin x}) \mathrm{~d} (\textcolor{lightgreen}{\arcsin x})
\end{align}
$$

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