From 22af3f17672f3d751ca7c51f7fb44ad41b9200a7 Mon Sep 17 00:00:00 2001
From: ViperEkura <3081035982@qq.com>
Date: Thu, 23 Apr 2026 15:17:22 +0800
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---
 subjects/math/04_积分.md | 30 ++++++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)

diff --git a/subjects/math/04_积分.md b/subjects/math/04_积分.md
index 53675ee..8f7300d 100644
--- a/subjects/math/04_积分.md
+++ b/subjects/math/04_积分.md
@@ -14,7 +14,37 @@ $$
 \int_a^b f(x)dx = \lim_{n\to\infty}\sum_{i=1}^n f(a + \frac{(b-a) i}{n} ) \frac{b-a}{n}
 $$
 
+### 要点 02 - sec x、csc x 的积分
+
+#### 基本积分公式
+
+$$\int \sec x \, dx = \ln|\sec x + \tan x| + C$$
+
+**推导方法**（分子分母策略）：
+
+$$\int \sec x \, dx = \int \sec x \cdot \frac{\sec x + \tan x}{\sec x + \tan x} \, dx = \int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, dx = \ln|\sec x + \tan x| + C$$
+
+---
+
+$$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$$
+
+**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
+
+$$\begin{aligned}
+\int \sec^2 x \, dx &= \tan x + C \\
+\int \csc^2 x \, dx &= -\cot x + C \\
+\int \sec x \tan x \, dx &= \sec x + C \\
+\int \csc x \cot x \, dx &= -\csc x + C \\
+\int \sec^3 x \, dx &= \frac{1}{2}(\sec x \tan x + \ln|\sec x + \tan x|) + C \\
+\int \csc^3 x \, dx &= \frac{1}{2}(-\csc x \cot x + \ln|\csc x + \cot x|) + C
+\end{aligned}$$
+
 ### 知识点
 - 定积分的定义
 - 黎曼和与积分的关系
 - 均匀分割技巧
+- sec x、csc x 的积分公式