From 9a6c53b1fcc6ca787712014c5d7eaf5a569fbb8e Mon Sep 17 00:00:00 2001
From: ViperEkura <3081035982@qq.com>
Date: Thu, 23 Apr 2026 15:11:06 +0800
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---
 subjects/math/02_导数与微分.md | 41 ++++++++++++++++++++++++++++++++++
 1 file changed, 41 insertions(+)

diff --git a/subjects/math/02_导数与微分.md b/subjects/math/02_导数与微分.md
index aa135a2..e18fab3 100644
--- a/subjects/math/02_导数与微分.md
+++ b/subjects/math/02_导数与微分.md
@@ -148,6 +148,45 @@ $$
 \end{aligned}
 $$
 
+### 要点 04 - sec x、csc x 的积分与转换关系
+
+#### 基本定义
+
+$$\sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}$$
+
+#### 常用恒等式（转换关系）
+
+$$\begin{aligned}
+&\sec^2 x = 1 + \tan^2 x \\
+&\csc^2 x = 1 + \cot^2 x \\
+&\sec x \cdot \csc x = \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x} \\
+&\sec^2 x - \tan^2 x = 1 \\
+&\csc^2 x - \cot^2 x = 1
+\end{aligned}$$
+
+#### 基本积分公式
+
+**推导方法**（分子分母策略）：
+
+$$\int \sec x \, dx = \int \sec x \cdot \frac{\sec x + \tan x}{\sec x + \tan x} \, dx = \int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, dx = \ln|\sec x + \tan x| + C$$
+
+---
+
+**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
+
+$$
+\begin{aligned}
+\int \sec^2 x \, dx &= \tan x + C \\
+\int \csc^2 x \, dx &= -\cot x + C \\
+\int \sec x \tan x \, dx &= \sec x + C \\
+\int \csc x \cot x \, dx &= -\csc x + C \\
+\end{aligned}
+$$
+
 ### 知识点
 - 莱布尼兹公式
 - 隐函数存在定理
@@ -155,3 +194,5 @@ $$
 - 参数方程求导
 - 曲率的定义
 - 曲率圆与曲率半径
+- sec x、csc x 的积分公式
+- 三角函数转换恒等式