From 1dd95d31377eb03fa66c417dd4650794e7ecc5ae Mon Sep 17 00:00:00 2001
From: InkSoul <qingci30@163.com>
Date: Mon, 25 Sep 2023 13:03:04 +0800
Subject: [PATCH] email update

---
 config.toml                     |  2 +-
 content/mathematics/公式大全.md | 23 +++++++++++++++++++++++
 2 files changed, 24 insertions(+), 1 deletion(-)

diff --git a/config.toml b/config.toml
index f0380e4..c52639f 100644
--- a/config.toml
+++ b/config.toml
@@ -98,7 +98,7 @@ uglyURLs = false
     # 名字
     name = "InkSoul"
     # 邮箱
-    email = "qingci30@163.com"
+    email = "blog@inksoul.top"
     # 座右铭或简介
     motto = ""
     # 头像
diff --git a/content/mathematics/公式大全.md b/content/mathematics/公式大全.md
index 07121a7..2da6a1c 100644
--- a/content/mathematics/公式大全.md
+++ b/content/mathematics/公式大全.md
@@ -53,6 +53,8 @@ date: 2022-12-24T13:05:40+08:00
           - [倒代换](#倒代换)
           - [指数代换](#指数代换)
           - [万能代换](#万能代换)
+        - [除法积分](#除法积分)
+        - [部分相消法](#部分相消法)
         - [组合积分法](#组合积分法)
       - [值得记忆的定积分公式](#值得记忆的定积分公式)
 - [线性代数](#线性代数)
@@ -399,6 +401,27 @@ $$
 
 原积分转化为$\int R(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})\frac{2}{1+t^2}dt$
 
+##### 除法积分
+
+$$
+\int \frac{u}{v}dx = \int \frac{u^{'}}{v^{'}}dx - \int (\frac{u}{v})^{'} \cdot \frac{u}{v^{'}} dx
+$$
+
+例如计算$\int \frac{x+\sin x}{1 + \cos x}dx$
+
+$原式 =  \int \frac{1+\cos x}{- \sin x}dx + \int (\frac{x+\sin x}{1+ \cos x})^{'} \cdot \frac{1+\cos x}{\sin x}dx \\\\=  \int \frac{1+\cos x}{-\sin x} + \int \frac{1+\cos x}{\sin x}d(\frac{x+\sin x}{1+\cos x}) \\\\ =\frac{1+\cos x}{\sin x} \cdot \frac{x+\sin x}{1 + \cos x} - \int \frac{x+\sin x}{1+ \cos x} \cdot \frac{-\sin^2 x -(1+\cos x) - x}{\sin^2 x}dx \\\\= \frac{x+\sin x}{\sin x}$
+
+##### 部分相消法
+
+对于一个可视为两个函数相乘的积分，可进行拆分后相消
+
+$原式 \rightarrow \\\\ \int f_1(x)dx + \int u(x)dv(x) \rightarrow \\\\ \int f_1(x)dx + u(x)v(x) - \int v(x)du(x) = \\\\ u(x)v(x)+C$
+
+例如计算$\int e^{2x}(\tan x + 1)^2dx$
+
+$原式 =  \int e^{2x}(\tan^2 x + 2\tan x +1)dx \\\\= \int e^{2x}(\sec^2 x + 2\tan x)dx \\\\= \int e^{2x}\sec^2 x dx + 2\int e^{2x}\tan x dx \\\\= \int e^{2x} d\tan x + 2\int e^{2x} \tan x dx \\\\= \tan x \cdot e^{2x} -2\int \tan x e^{2x} dx + 2\int e^{2x} \tan x  dx \\\\ = \tan x \cdot e^{2x} + C
+$
+
 ##### 组合积分法
 
 对于